Modulo orientations with bounded out-degrees

TL;DR

This paper proves that highly edge-connected graphs and graphs with many spanning trees can be oriented to have balanced out-degrees within certain bounds, extending understanding of graph orientations with degree constraints.

Contribution

It establishes new conditions under which graphs can be oriented to achieve bounded out-degrees, linking edge-connectivity and spanning trees to orientation properties.

Findings

Graphs with (3k-3)-edge-connectivity admit orientations with bounded out-degree deviations.

Graphs with 2k-2 edge-disjoint spanning trees can also be oriented with bounded out-degree, with stricter bounds.

The results generalize and strengthen previous orientation theorems based on connectivity and spanning tree conditions.

Abstract

Let $G$ be a graph, let $k$ be a positive integer, and let $p:V(G)\rightarrow Z_k$ be a mapping with $|E(G)| \stackrel{k}{\equiv}\sum_{v\in V(G)}p(v) $. In this paper, we show that if $G$ is $(3k-3)$-edge-connected, then it has an orientation such that for each vertex $v$, $|d^+_G(v)-d_G(v)/2| < k$; also if $G$ contains $2k-2$ edge-disjoint spanning trees, then it admits such an orientation but by imposing greater out-degree bounds.

Equations

$$k-1\le d_G^{+}(v)\le \max \{2k-2,d_G(v)-(k-1)\}.$$

$$\omega (G\setminus (A\cup B))\le 1+\sum _{v\in A}q(v)-e_G(A)+\sum _{v\in B}(d_G(v)-p(v))-e_G(B).$$

$$\lfloor \frac{d_G(v)}{2}\rfloor -1\le d_G^{+}(v)\le \lceil \frac{d_G(v)}{2}\rceil +1.$$

$$\omega (G\setminus (A\cup B))\le \sum _{A\cup B}\frac{1}{2}{d}_G(v)-e_G(A\cup B),$$

$$\omega (G\setminus (A\cup B))\le 3/2+\sum _{v\in A}{q}^{\prime }(v)-e_G(A)+\sum _{v\in B}(d_G(v)-p^{\prime }(v))-e_G(B).$$

$$s(v)\le d_G^{+}(v)\le d_G(v)-s_0(v).$$

$$\omega (G\setminus (A\cup B))\le \omega (T\setminus (A\cup B))\le \sum _{A\cup B}(d_T(v)-1)+1-e_T(A\cup B).$$

$$0\le \sum _{v\in A\cup B}{d}_F^{-}(v)-e_F(A\cup B)\le \sum _{v\in A}(d_F(v)-s_0(v))+\sum _{v\in B}(d_F(v)-s(v))-e_F(A\cup B).$$

$$\omega (G\setminus (A\cup B))\le \sum _{v\in A}(d_G(v)-s_0(v)-1)+\sum _{v\in B}(d_G(v)-s(v)-1)-e_G(A\cup B)+1,$$

$$\omega (G\setminus (A\cup B))\le 1+\sum _{v\in A}{q}^{\prime }(v)-e_G(A)+\sum _{v\in B}(d_G(v)-p^{\prime }(v))-e_G(B).$$

$$p^{\prime }=\{\begin{array}{ll}p-{\chi }_u-{\chi }_x,& \text{if xu and yu are parallel};\\ p-{\chi }_u,& \text{if xu and yu are not parallel}.\end{array}$$

$$|d_G^{+}(v)-d_G(v)/2|\le k-1+|\alpha (v)|.$$

$$|\alpha (W)|=|\alpha (\{z_0,v_0\})|=|\alpha (z_0)|,$$

$$d_G(z_0)=d_G(W)+d_G(v_0)\ge (2k-2)+2|\alpha (W)|+2=2k+2|\alpha (z_0)|.$$

$$d_G(x)=d_G(\{x,v_0\})+d_G(v_0)\ge (2k-2)+2|\alpha (\{x,v_0\})|+2=2k+2|\alpha (x)|.$$

$$d_G(z_0)\ge d_G(U)+d_G(W)>(2k-2)+k\ge (2k-2)+2|\alpha (z_0)|,$$

$$d_G(y)\ge \{\begin{array}{ll}{d}_G(x)+d_G(z_0)\ge 2k+k\ge 2k+2|\alpha (y)|,& \text{if ∣V(G)∣=3};\\ d_G(x)+d_G(\{x,y\})\ge 2k+2k-2\ge 2k+2|\alpha (y)|,& \text{if ∣V(G)∣>3}.\end{array}$$

$$|d_G^{+}(v)-d_G(v)/2|\le k-1+|\alpha (v)|.$$

$$d_{G^{\prime }}(x)=$$

$$d_{G^{\prime }}(y)=$$

$$d_{G^{\prime}}(A)=d_{G}(A)-1\geq\big{(}2k+2|\alpha(A)|\big{)}-1\geq(2k-2)+2|\alpha^{\prime}(A)|.$$

$$d_G(x)\ge (2k-2)+2\alpha (x)\text{ and }0<\alpha (x)<k/2.$$

$$p(z_0)\stackrel{k}{\equiv }\alpha (z_0)+d_G(z_0)/2\stackrel{k}{\equiv }\alpha (z_0)+(2k+p(z_0))/2\stackrel{k}{\equiv }\alpha (z_0)+p(z_0)/2,$$

$${\alpha }^{\prime }(z_0)\stackrel{k}{\equiv }{p}^{\prime }(z_0)-d_{G^{\prime }}(z_0)/2\stackrel{k}{\equiv }p(z_0)-1-(d_G(z_0)+k-2)/2\stackrel{k}{\equiv }p(z_0)/2.$$

$$d_{G^{\prime }}(x)=d_G(x)+(k-2)\ge (2k-2)+2|\alpha (x)|+(k-2)\ge 3k-3,$$

$$\alpha^{\prime}(x)\stackrel{{\scriptstyle k}}{{\equiv}}p^{\prime}(x)-d_{G^{\prime}}(x)/2\stackrel{{\scriptstyle k}}{{\equiv}}p(x)-1-\big{(}d_{G}(x)+(k-2)\big{)}/2\stackrel{{\scriptstyle k}}{{\equiv}}\alpha(x)-k/2.$$

$$d_{G^{\prime }}(A)=d_G(A)+k-2\ge 3k-2=(2k-2)+k\ge (2k-2)+2|{\alpha }^{\prime }(A)|.$$

$$|d_G^{+}(v)-d_G(v)/2|\le k-1+|\alpha (v)|.$$

$$d_G(y)\ge d_G(z)+d_G(\{z,y\})\ge 2k+2k-2\ge 2k+2|\alpha (y)|.$$

$$\lfloor \frac{d_G(v)}{2}\rfloor -(k-1)\le d_G^{+}(v)\le \lceil \frac{d_G(v)}{2}\rceil +(k-1).$$

$$|d_G^{+}(v)-d_G(v)/2|\le k-1+|\alpha (v)|.$$

Full text

*Keywords: Modulo orientation; out-degree; edge-connected; partition-connected. *

1 Introduction

In this article, graphs have no loops, but multiple edges are allowed, and a general graph may have loops and multiple edges. Let $G$ be a graph. The vertex set and the edge set of $G$ are denoted by $V(G)$ and $E(G)$, respectively. We denote by $d_{G}(v)$ the degree of a vertex $v$ in the graph $G$, whether $G$ is directed or not. If $G$ has an orientation, the out-degree and in-degree of $v$ are denoted by $d_{G}^{+}(v)$ and $d_{G}^{-}(v)$. For a vertex set $A$ of $G$ with at least two vertices, the number of edges of $G$ with exactly one end in $A$ is denoted by $d_{G}(A)$. Also, we denote by $e_{G}(A)$ the number of edges with both ends in $A$ and denote by $d_{G}(A,B)$ the number of edges with one end in $A$ and one end in $B$, where $B$ is a vertex set. For notational simplicity, we write $A^{c}$ for the vertex set $V(G)\setminus A$. We denote by $G[A]$ the induced subgraph of $G$ with the vertex set $A$ containing precisely those edges of $G$ whose ends lie in $A$, and denote by $G[A,B]$ the induced bipartite factor of $G$ with the bipartition $(A,B)$. The graph $G$ is said to be trivial, if it has no edges. Let $k$ be a positive integer. The cyclic group of order $k$ is denoted by $Z_{k}$. For any integer $n$, we denote by $[n]_{k}^{-}$ the unique integer $n_{0}$ such that $n_{0}\stackrel{{\scriptstyle k}}{{\equiv}}n$ and $n_{0}\in\{-1,0,\ldots,k-2\}$. For convenience, we write $[n]_{k}$ for $[n]^{-}_{k}$. An orientation of $G$ is said to be $p$-orientation, if for each vertex $v$, $d_{G}^{+}(v)\stackrel{{\scriptstyle k}}{{\equiv}}p(v)$, where $p:V(G)\rightarrow Z_{k}$ is a mapping. For two rational numbers $x$ and $y$, we say that $x\stackrel{{\scriptstyle k}}{{\equiv}}y$, if $x-y$ is an integer divisible by $k$. A graph $G$ is called $m$-tree-connected, if it contains $m$ edge-disjoint spanning trees. Note that by the result of Nash-Williams [11] and Tutte [14] every $2m$-edge-connected graph is $m$-tree-connected. A graph $G$ is said to be $(m,l_{0})$-partition-connected, if it can be decomposed into an $m$-tree-connected factor and a factor $F$ having an orientation such that for each vertex $v$, $d^{+}_{F}(v)\geq l_{0}(v)$, where $l_{0}$ is an integer-valued function on $V(G)$. A graph is termed essentially $\lambda$-edge-connected, if all edges of any edge cut of size strictly less than $\lambda$ are incident with a common vertex. A graph $G$ is called odd-$\lambda$-edge-connected, if $d_{G}(A)\geq\lambda$ for every vertex set $A$ with $d_{G}(A)$ odd. For a graph $G$ with a given vertex $u$, we denote by $\chi_{u}$ the mapping $\chi_{u}:V(G)\rightarrow\{0,1\}$ such that $\chi_{u}(u)=1$ and $\chi_{u}(v)=0$ for all vertices $v$ with $v\neq u$. Two different edges are called parallel, if have the same end vertices. For two edges $xu$ and $uy$ incident with the vertex $u$, lifting of $xu$ and $uy$ is an operation that removes $xu$ and $uy$ and adds a new edge $xy$ (when the purpose is to generate a loopless graph we must not add the next edge $xy$ when $x=y$). Also, if one of $xu$ and $yu$, as $xu$, is directed, then we direct $xy$ toward $y$ when $xu$ is toward $u$, and direct $xy$ away from $y$ when $xu$ is away from $u$. Throughout this article, all variables $m$ are nonnegative integers and all variables $k$ are positive integers.

In 2012 Thomassen constructed the following theorem about the existence of modulo orientations in highly edge-connected graphs.

Theorem 1.1

.([12])* Let $G$ be a graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(2k^{2}+k)$-edge-connected, then it has a $p$-orientation.*

Later, Lovász, Thomassen, Wu, and Zhang (2013) refined Theorem 1.1 for odd integers $k$ by reducing the quadratic bound $(2k^{2}+k)$ down to a linear bound as the following theorem. They also remarked that this number can be reduced to $3k-2$ for even integers $k$.

Theorem 1.2

.([9])* Let $G$ be a graph, let $k$ be an odd integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(3k-3)$-edge-connected, then $G$ has a $p$-orientation. In particular, the needed edge-connectivity can be replaced by odd-edge-connectivity $3k-2$, when $2p(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}(v)$ for all vertices $v$.*

In this paper, we refine Theorem 1.2 by pushing the required edge-connectivity down to $3k-3$, even for even numbers $k$, and strengthen it by giving a sharp bound on out-degrees as mentioned in the abstract. In addition, we improve the needed edge-connectivity for Eulerian graphs for the following special case.

Theorem 1.3

.* Let $k$ be an even positive integer, let $G$ be an Eulerian graph, and let $Q\subseteq V(G)$ with $|Q|$ even. If $d_{G}(X)\geq 3k-2$ for every $X\subseteq V(G)$ with $|X\cap Q|$ odd, then $G$ admits an orientation such that for each $v\in V(G)\setminus Q$, $d_{G}^{+}(v)=d_{G}^{-}(v)$, and for each $v\in Q$, $|d_{G}^{+}(v)-d_{G}^{-}(v)|=k$.*

Recently, Thomassen (2020) applied Theorems 1.1 and 1.2 to establish the following elegant result about the existence of regular factorizations of edge-connected regular graphs. As an application, we applied these new improvements in [7] to refine Thomassen’s result.

Theorem 1.4

.([13])* Let $r$ be a natural number, $r\geq 4$ and let $G$ be an $r$-regular graph. Let $r=kq$ be a natural number where $q\geq 3$ is odd. If $k$ is odd and $G$ has odd-edge-connectivity at least $3k-2$, then $G$ can be edge-decomposed into $q$-factors. If $k$ is even and $G$ has an even number of vertices and edge-connectivity at least $k^{2}+2k$, then $G$ can be edge-decomposed into $q$-factors.*

In Section 4, we investigate modulo orientations with bounded out-degrees in partition-connected graphs and strengthen some recent results in [6, 8] toward this concept. As a consequence, we prove the following theorem.

Theorem 1.5

.* Let $G$ be a graph with $z\in V(G)$, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(2k-2)$-tree-connected, then it has a $p$-orientation such that for each $v\in V(G)\setminus\{z\}$,*

[TABLE]

2 Orientations modulo $2$

In this section, we consider the existence of parity orientations. Our results are based on the following theorem which is a special case of a result due to Frank, Tardos, and Sebő (1984) who gave a criterion for the existence of parity factors with bounded degrees.

Theorem 2.1

.(see Theorem 6 in [4])* Let $G$ be a connected graph and let $p$ and $q$ be two integer-valued functions on $V(G)$ with $p\leq q$ satisfying $|E(G)|\stackrel{{\scriptstyle 2}}{{\equiv}}\sum_{v\in V(G)}p(v)$ and $p(v)\stackrel{{\scriptstyle 2}}{{\equiv}}q(v)$ for each vertex $v$. Then $G$ has a $p$-orientation modulo $2$ such that for each vertex $v$, $p(v)\leq d^{+}_{G}(v)\leq q(v)$, if for any two disjoint subsets $A$ and $B$ of $V(G)$ with $A\cup B\neq\emptyset$,*

[TABLE]

Corollary 2.2

.(see Theorem 4 in [4])* Let $G$ be a graph and let $p:V(G)\rightarrow Z_{2}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle 2}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is connected, then $G$ has a $p$-orientation.*

2.1 $2$-edge-connected graphs

As we have stated above, edge-connectedness $1$ is sufficient for a graph to have a $p$-orientation modulo $2$. Here, we show that edge-connectedness $2$ is sufficient for a graph to have a $p$-orientation modulo $2$ in which out-degrees fall in predetermined short intervals.

Theorem 2.3

.* Let $G$ be a graph and let $p:V(G)\rightarrow Z_{2}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle 2}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $2$-edge-connected, then it has a $p$-orientation such that for each vertex $v$,*

[TABLE]

Furthermore, for an arbitrary vertex $z$, $d^{+}_{G}(z)$ can be assigned to any plausible integer value in whose interval.

**Proof. **

For each vertex $v$, define $p^{\prime}(v)\in\{\lfloor d_{G}(v)/2\rfloor-1,\lfloor d_{G}(v)/2\rfloor\}$ and $q^{\prime}(v)\in\{\lceil d_{G}(v)/2\rceil,\lceil d_{G}(v)/2\rceil+1\}$ such that $p^{\prime}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}p(v)\stackrel{{\scriptstyle 2}}{{\equiv}}q^{\prime}(v)$. Obviously, $p^{\prime}(v)\leq q^{\prime}(v)$. Note that if $p^{\prime}(z)<q^{\prime}(z)$, we permit to replace $p^{\prime}(z)$ by $p^{\prime}(z)+2$ or replace $q^{\prime}(z)$ by $q^{\prime}(z)-2$ with respect to our purpose related to $z$. For the first option, we have $p^{\prime}(z)\leq d_{G}(z)/2+3/2$ and for the second option, we have $d_{G}(z)/2\leq q^{\prime}(z)+3/2$. Let $A$ and $B$ be two disjoint subsets of $V(G)$ with $A\cup B\neq\emptyset$. Since $G$ is $2$-edge-connected, it is easy to see that

[TABLE]

which implies that

[TABLE]

Thus by Theorem 2.1, the graph $G$ has a $p$-orientation such that for each vertex $v$, $p^{\prime}(v)\leq d^{+}_{G}(v)\leq q^{\prime}(v)$, and the proof is completed.

• $\Box$*

We shall here introduce an alternative proof using an induction.

**Proof. **

By induction on the sum of all $d_{G}(v)-3$ taken over all vertices $v$ with $d_{G}(v)\geq 4$. First, assume that for each vertex $v$, $d_{G}(v)\leq 3$. For $|V(G)|=1$, the proof is clear. So, suppose $|V(G)|\geq 2$ which implies that $G$ has no loops. It is not hard to check that there is an edge set $E$ incident with $z$ such that $G-E$ is connected, where $|E|=1$ when $z$ has even degree and $|E|=2$ when $z$ has odd degree. Orient the edge(s) of $E$ toward $z$, if the desired condition on out-degree of $z$ is $d^{+}_{G}(z)\leq\lfloor d_{G}(z)/2\rfloor$, and orient the edge(s) of $E$ away from $z$ if the desired condition on out-degree of $z$ is $d^{+}_{G}(z)\geq\lceil d_{G}(z)/2\rceil$. By applying Corollary 2.2 to the graph $G-E$, the pre-orientation of $E$ can be extended to a $p$-orientation of $G$ satisfying the theorem. Now, assume that for a vertex $u$, $d_{G}(u)\geq 4$. By Fleischner’s splitting lemma, there are two edges $xu$ and $yu$ of $G$ incident with $u$ such that by lifting them the resulting general graph $H$ is still $2$-edge-connected (possibly $x=u$ or $y=u$). Note that for each vertex $v$, $d_{H}(v)=d_{G}(v)-2\chi_{u}$. Define $p^{\prime}=p-\chi_{u}$ so that $|E(H)|\stackrel{{\scriptstyle 2}}{{\equiv}}\sum_{v\in V(H)}p^{\prime}(v)$. By the induction hypothesis, $H$ admits a $p^{\prime}$-orientation such that for each vertex $v$, $\lfloor d_{H}(v)/2\rfloor-1\leq d_{H}^{+}(v)\leq\lceil d_{H}(v)/2\rceil+1$. This orientation of $H$ induces an orientation for $G$ such that for each vertex $v$, $d_{G}^{+}(v)=d^{+}_{H}(v)+\chi_{u}$. This orientation of $G$ is a $p$-orientation satisfying the desired properties. The extra condition on $d^{+}_{G}(z)$ can be obtained by giving an appropriate condition on $d^{+}_{H}(z)$. Hence the theorem holds.

• $\Box$*

2.2 $(1,l_{0})$-partition-connected graphs

In the following theorem, we develop Theorem 2.3 to a partition-connected version.

Theorem 2.4

.* Let $G$ be a graph and let $p:V(G)\rightarrow Z_{2}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle 2}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Let $s$, $s_{0}$, and $l_{0}$ be three integer-valued functions on $V(G)$ satisfying $s+s_{0}<d_{G}$ and $\max\{s,s_{0}\}\leq l_{0}$. If $G$ is $(1,l_{0})$-partition-connected, then it has a $p$-orientation such that for each vertex $v$,*

[TABLE]

**Proof. **

For each vertex $v$, define $p^{\prime}(v)\in\{s(v),s(v)+1\}$ and $q^{\prime}(v)\in\{d_{G}(v)-s_{0}(v)-1,d_{G}(v)-s_{0}(v)\}$ such that $p^{\prime}(v)\stackrel{{\scriptstyle 2}}{{\equiv}}p(v)\stackrel{{\scriptstyle 2}}{{\equiv}}q^{\prime}(v)$. Note that the condition $s(v)+s_{0}(v)<d_{G}(v)$ implies that $p^{\prime}(v)\leq q^{\prime}(v)+1$ and hence $p^{\prime}(v)\leq q^{\prime}(v)$. By the assumption, the graph $G$ can be decomposed into two factors $T$ and $F$ such that $T$ is a spanning tree and $F$ admits an orientation such that for each vertex $v$, $d^{+}_{F}(v)\geq l_{0}(v)$. Let $A$ and $B$ be two disjoint subsets of $V(G)$. Since $T$ is connected, it is easy to see that

[TABLE]

Moreover, since $d^{+}_{F}(v)\geq l_{0}(v)\geq\max\{s(v),s_{0}(v)\}$ for each vertex $v$, we must have

[TABLE]

Therefore,

[TABLE]

which implies that

[TABLE]

Thus by Theorem 2.1, the graph $G$ has a $p$-orientation such that for each vertex $v$, $p^{\prime}(v)\leq d^{+}_{G}(v)\leq q^{\prime}(v)$, and the proof is completed.

• $\Box$*

3 Orientations modulo $k$: edge-connected graphs

In this section, we are going to improve Theorem 1.2 by giving a sharp bound on out-degrees and provide a common version for odd and even integers $k$. We follow with the same innovative ideas that appeared in [9] and retain the same arguments, while modifications are inserted. The proof is based on defining a set function $\alpha$ whose values lie in the set $\{0,\pm 1/2,\ldots,\pm k/2\}$. It is inspired by the set function $\tau(A)$ in [9] and the set function $t(A)$ in [12]. More precisely, for odd integers $k$, $2\alpha(A)=\tau(A)$, and for odd and even integers $k$, $2|\alpha(A)|=t(A)$.

3.1 Definition and properties of set functions $\alpha$

Let $G$ be a graph, let $k$ be a positive integer, and let $p:V(G)\rightarrow Z_{k}$ be a mapping. For each vertex $v$, take $\alpha(v)$ to be a rational number such that $\alpha(v)\in\{0,\pm 1/2,\ldots,\pm k/2\}$ and $\alpha(v)\stackrel{{\scriptstyle k}}{{\equiv}}p(v)-d_{G}(v)/2$. In intuitive terms, $|\alpha(v)|$ specifies the distance between two points $p(v)$ and $d_{G}(v)/2$ on a circle whose circumference is $k$, and the sign of $\alpha(v)$ determines the position of $p(v)$ with respect to $d_{G}(v)/2$. Thus, it is intuitively clear and not difficult to show that $\alpha(v)$ is unique unless $\alpha(v)\in\{-k/2,k/2\}$. For any vertex set $A$, take $\alpha(A)$ to be a rational number such that $\alpha(A)\in\{0,\pm 1/2,\ldots,\pm k/2\}$ and $\alpha(A)\stackrel{{\scriptstyle k}}{{\equiv}}p(A)-d_{G}(A)/2,$ where $p(A)=\sum_{v\in A}p(v)\,-e_{G}(A)$ and $d_{G}(A)=\sum_{v\in A}d_{G}(v)\,-2e_{G}(A)$. When $G$ and $p$ are not clear from the context, we denote by $\alpha_{G}(v,p)$ and $\alpha_{G}(A,p)$ the value of $\alpha(v)$ and $\alpha(A)$. Now, we present some basic properties of $\alpha$ in the following propositions.

Proposition 3.1

.* Let $G$ be a graph and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. For any two vertex sets $A$ and $B$, the following statements hold:*

If $\alpha(A)\stackrel{{\scriptstyle k}}{{\equiv}}\pm\alpha(B)$, then $|\alpha(A)|=|\alpha(B)|$. 2. 2.

If $A\cap B=\emptyset$, then $\alpha(A\cup B)\stackrel{{\scriptstyle k}}{{\equiv}}\alpha(A)+\alpha(B).$ 3. 3.

$|\alpha(A)|=|\alpha(A^{c})|.$** 4. 4.

If $\alpha(v_{0})=0$ for a vertex $v_{0}$ with $v_{0}\in V(G)\setminus A$, then $|\alpha(A)|=|\alpha(A\cup\{v_{0}\})|.$ 5. 5.

If $d_{G}(A)\geq 3k-3$, then $d_{G}(A)\geq(2k-2)+2|\alpha(A)|$. 6. 6.

$d_{G}(A)-2|\alpha(A)|$* is an even integer.*

**Proof. **

To obtain (1), one can conclude that $|\alpha(A)\mp\alpha(B)|\in\{0,k\}$ which implies that $|\alpha(A)|=|\alpha(B)|$. To prove (2), it suffices to check that \alpha(A)+\alpha(B)\stackrel{{\scriptstyle k}}{{\equiv}}(p(A)-d_{G}(A)/2)+(p(B)-d_{G}(B)/2)\stackrel{{\scriptstyle k}}{{\equiv}}\big{(}p(A)+p(B)-d_{G}(A,B)\big{)}-\big{(}d_{G}(A)+d_{G}(B)-2d_{G}(A,B)\big{)}/2 which implies that $\alpha(A)+\alpha(B)\stackrel{{\scriptstyle k}}{{\equiv}}p(A\cup B)-d_{G}(A\cup B)/2\stackrel{{\scriptstyle k}}{{\equiv}}\alpha(A\cup B).$ Moreover, \alpha\big{(}V(G)\big{)}=0, since p\big{(}V(G)\big{)}\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)\;-e_{G}(V(G))\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)\;-|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}0. Hence $\alpha(A)+\alpha(A^{c})\stackrel{{\scriptstyle k}}{{\equiv}}0$ and $|\alpha(A)|=|\alpha(A^{c})|$ which establishes (3). The proof of (4) can be obtained from $\alpha(A)\stackrel{{\scriptstyle k}}{{\equiv}}\alpha(A)+\alpha(v_{0})\stackrel{{\scriptstyle k}}{{\equiv}}\alpha(A\cup\{v_{0}\})$. Since $\alpha(A)+d_{G}(A)/2$ is an integer, $2\alpha(A)+d_{G}(A)$ is even and so $d_{G}(A)-2|\alpha(A)|$ is even which implies (6). Note that $|\alpha(A)|\leq k/2$. If $|\alpha(A)|=k/2$, then $d_{G}(A)$ and $k$ have the same parity. Since $3k-3$ and $k$ have different parity, we have $|\alpha(A)|<k/2$, when $d_{G}(A)=3k-3$. This can complete the proof.

• $\Box$*

Proposition 3.2

.* Let $G$ be a graph and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G^{\prime}$ is a graph obtained from $G$ by lifting two edges $xu$ and $yu$, then for every vertex set $A$ we have $|\alpha^{\prime}(A)|=|\alpha(A)|$, where*

[TABLE]

3.2 Graphs with edge-connectivity at least $3k-3$

Now, we are ready to refine the main result in [9].

Theorem 3.3

.* Let $G$ be a graph with $z_{0}\in V(G)$, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Let $D_{z_{0}}$ be a pre-orientation of $E(z_{0})$ that is the set of edges incident with $z_{0}$. Let $V_{0}=\{v\in V(G)-z_{0}:\alpha(v)=0\}$. If $V_{0}\neq\emptyset$, we let $v_{0}$ be a vertex of $V_{0}$ with smallest degree. Assume that*

1. $(i)$

$d_{G}(z_{0})\leq 2k-2+2|\alpha(z_{0})|$, and the edges incident with $z_{0}$ are pre-directed such that $d^{+}_{G}(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}p(z_{0})$. 2. $(ii)$

$d_{G}(A)\geq 2k-2+2|\alpha(A)|$, for any vertex set $A$ with $\emptyset\subsetneq A\subsetneq V(G)\setminus\{z_{0}\}$ and $A\neq\{v_{0}\}$.

Then the pre-orientation $D_{z_{0}}$ can be extended to a $p$-orientation $D$ of $G$ such that for each vertex $v$,

[TABLE]

**Proof. **

The proof is by contradiction. We assume (reductio ad absurdum) that $(G,p,z_{0})$ is a counterexample so that $|V(G)|\geq 3$. That is, the graph $G$ with mapping $p$ satisfies the conditions of the theorem but some pre-orientation $D_{z_{0}}$ cannot be extended to a $p$-orientation of $G$ with the desired properties. Let $\mathcal{M}$ be the collection of counterexamples $(G,p,z_{0})$ such that $|V(G)|+|E(G-z_{0})|$ is minimum. The proof is divided into two parts. The first part, Claims 1-5 below, establishes some properties of all members of $\mathcal{M}$. In the second part we choose a member $(G,p,z_{0})$ of $\mathcal{M}$ such that $|E(G)|$ is minimum and prove that it is not a counterexample, yielding a contradiction. If we work with distinct graphs $G$, $G^{\prime}$, we use the terms $p(A)$ and $\alpha(A)$ when $A$ is a vertex set of $G$, and $p^{\prime}(A)$ and $\alpha^{\prime}(A)$ when $A$ is a vertex set of $G^{\prime}$.

Part I. Some properties of $\mathcal{M}$.

In Part 1 we let $(G,p,z_{0})$ be any member of $\mathcal{M}$.

Claim 1. For every vertex set $A\subsetneq V(G)\setminus\{z_{0}\}$ with $|A|\geq 2$, we have $d_{G}(A)\geq 2k+2|\alpha(A)|$.

If $d_{G}(A)<2k+2|\alpha(A)|$, then we first get an extension of $D_{z_{0}}$ to the contracted graph $H=G/A$ by the minimality property of $G$, since $|V(H)|<|V(G)|$ and $|E(H-z_{0})|\leq|E(G-z_{0})|$. Then all edges of the edge-cut $[A,A^{c}]$ are oriented in this extension, where $A^{c}=V(G)\setminus A$ and $[A,A^{c}]$ is the set of edges with exactly one end in $A$. Similarly we then contract $A^{c}$ into a single vertex as a new $z_{0}$, and again, we use the minimality of $G$ to extend the orientation of $[A,A^{c}]$ to the edges of $G$ with both ends in $A$. $\square$

Claim 2. $V_{0}=\emptyset$.

Suppose $V_{0}\neq\emptyset$ and $v_{0}$ is a vertex of $V_{0}$ with smallest degree. We can assume that $d_{G}(v_{0})\geq 2$. Otherwise $v_{0}$ is an isolated vertex and we can remove it and use the minimality of $G$. If $v_{0}$ has at least two neighbours, we lift one pair of edges incident with $v_{0}$ which are not parallel. Claim 1 implies that the resulting graph $G^{\prime}$ with the modified mapping $p^{\prime}=p-\chi_{v_{0}}$ satisfies the hypotheses of the theorem. Since $|E(G^{\prime}-z_{0})|<|E(G-z_{0})|$, it holds that $G^{\prime}$ has the desired orientation, and so does $G$, a contradiction.

Now suppose $v_{0}$ has only one neighbour $x$. We must have $x\neq z_{0}$. Otherwise,

[TABLE]

where $W=V(G)\setminus\{z_{0},v_{0}\}$, and then

[TABLE]

a contradiction to condition (i). If $|V(G)|=3$, then we extend $D_{z_{0}}$ to an orientation of $G$ by orienting half of the edges between $x$ and $v_{0}$ toward $v_{0}$ and the other half away from $v_{0}$, yielding a contradiction. For the case $|V(G)|>3$, we have

[TABLE]

Then, we lift one pair of edges incident with $v_{0}$ and $x$ which are parallel. Claim 1 implies that the resulting graph $G^{\prime}$ with the modified mapping $p^{\prime}=p-\chi_{x}-\chi_{v_{0}}$ satisfies the hypotheses of the theorem. Since $|E(G^{\prime}-z_{0})|<|E(G-z_{0})|$, it holds that $G^{\prime}$ has the desired orientation, and so does $G$, again a contradiction. $\square$

Claim 3. $G-z_{0}$ is connected, and $d_{G}(z_{0})\geq k$.

Suppose $G-z_{0}$ is disconnected and let $U$ and $W$ be two components of $G-z_{0}$. By condition (ii) and Claim 2, we have $d_{G}(U)\geq 2k-2$ and $d_{G}(W)\geq 2k-2$. Then

[TABLE]

a contradiction to condition (i).

Suppose $d_{G}(z_{0})\leq k-1$ and let $G^{\prime}$ be the graph constructed from $G$ by replacing an edge $xy$ of $G-z_{0}$ with a directed path of length two through $z_{0}$ with $p^{\prime}=p+\chi_{z_{0}}$. We have $d_{G^{\prime}}(z_{0})\leq(k-1)+2\leq 2k-2\leq(2k-2)+2|\alpha^{\prime}(z_{0})|$ and hence $G^{\prime}$ satisfies condition (i). For any vertex set $A$ described in condition (ii), $d_{G^{\prime}}(A)=d_{G}(A)+2$, if $A$ contains both $x$ and $y$, and $d_{G^{\prime}}(A)=d_{G}(A)$ otherwise. So condition (ii) is clearly satisfied. Since $|V(G^{\prime})|=|V(G)|$ and $|E(G^{\prime}-z_{0})|<|E(G-z_{0})|$, this implies (by the definition of $\mathcal{M}$) that an extension of $D_{z_{0}}$ exists in $G^{\prime}$. This orientation results in an orientation of $G$, a contradiction. $\square$

Claim 3.A. For each vertex $v\in V(G)-z_{0}$, $d_{G}(v)=2k-2+2|\alpha(v)|$.

Suppose otherwise that $d_{G}(x)\geq 2k+2|\alpha(x)|$ for a vertex $x$ with $x\neq z_{0}$. First, assume that $x$ has at least two neighbours. Then we lift one pair of edges incident with $x$ which are not parallel. Claim 1 implies that the resulting graph $G^{\prime}$ with the modified mapping $p^{\prime}=p-\chi_{x}$ satisfies the hypotheses of the theorem. Since $|E(G^{\prime}-z_{0})|<|E(G-z_{0})|$, it holds that $G^{\prime}$ has the desired orientation, and so does $G$, a contradiction. Next, assume that $x$ has only one neighbour $y$. By Claim 3, we must have $y\neq z_{0}$ and so

[TABLE]

Then, we lift one pair of edges incident with $x$ and $y$ which are parallel. Claim 1 implies that the resulting graph $G^{\prime}$ with the modified mapping $p^{\prime}=p-\chi_{x}-\chi_{y}$ satisfies the hypotheses of the theorem. Since $|E(G^{\prime}-z_{0})|<|E(G-z_{0})|$, it holds that $G^{\prime}$ has the desired orientation, and so does $G$, again a contradiction. $\square$

By condition (i) and Claim 3.A, if $G$ has a $p$-orientation, then for each vertex $v$ the following condition automatically holds,

[TABLE]

Claim 4. For any two distinct vertices $x,y\in V(G)-z_{0}$, we have $\alpha(x)\alpha(y)>0$.

Suppose $\alpha(x)\alpha(y)\leq 0$. By Claim 2, we may assume that $\alpha(x)>0$ and $\alpha(y)<0$. By Claim 3, since $G-z_{0}$ is connected, we may also assume that $xy\in E(G-z_{0})$. Let $G^{\prime}=G-xy$, and take $p^{\prime}=p-\chi_{x}$ to be the modified mapping.

Then $|V(G^{\prime})|=|V(G)|$ and $|E(G^{\prime}-z_{0})|<|E(G-z_{0})|$. If $G^{\prime}$ and $p^{\prime}$ satisfy the conditions of the theorem, then by the definition of $\mathcal{M}$, the pre-orientation can be extended to a $p^{\prime}$-orientation of $G^{\prime}$ and further to a $p$-orientation of $G$ by adding a directed edge from $x$ to $y$, yielding a contradiction. Hence, it suffices to verify the conditions of the theorem for $G^{\prime}$ and $p^{\prime}$. Moreover, we only need to verify condition (ii) for single vertices $x$ and $y$ and vertex sets $A$ such that $|A|\geq 2$ and $d_{G^{\prime}}(A)=d_{G}(A)-1$ which are affected by the deletion of $xy$.

Condition (ii) is satisfied for $x$ and $y$, since

[TABLE]

For any vertex set $A$ (in condition (ii)) such that $|A|\geq 2$ and $d_{G^{\prime}}(A)=d_{G}(A)-1$, we have $|\alpha^{\prime}(A)|=|\alpha(A)\pm 1/2|\leq|\alpha(A)|+1/2$ and by Claim 1,

[TABLE]

Hence condition (ii) is verified for $A$. So $\alpha(x)\alpha(y)>0$. $\square$

Let $V^{+}=\{x\in V(G)-z_{0}:0<\alpha(x)<k/2\}$ and $V^{-}=\{x\in V(G)-z_{0}:-k/2<\alpha(x)<0\}$.

Note that if $k/2\stackrel{{\scriptstyle k}}{{\equiv}}p(x)-d_{G}(x)/2$, then $\alpha(x)$ has two possible values, namely $k/2$ and $-k/2$.

Claim 5 . $V(G)-z_{0}=V^{+}$ or $V(G)-z_{0}=V^{-}$.

By Claim 4, we have $V^{+}=\emptyset$ or $V^{-}=\emptyset$. So it suffices to prove that $|\alpha(x)|<k/2$ for any vertex $x$ other than $z_{0}$. If $x\in V(G)-z_{0}$ such that $|\alpha(x)|=k/2$, then for any vertex $y$ distinct from $x$ and $z_{0}$, we can choose $\alpha(x)=k/2$ or $\alpha(x)=-k/2$ such that $\alpha(x)\alpha(y)\leq 0$ and get a contradiction to Claim 4. $\square$

Part II. Minimum members of $\mathcal{M}$.

Now choose $(G,p,z_{0})$ to be a member of $\mathcal{M}$ such that $|E(G)|$ is minimum. Without loss of generality, assume that $V(G)-z_{0}=V^{+}$. For if $V(G)-z_{0}=V^{-}$, we reverse the directions of all edges incident with $z_{0}$ and replace $p(x)$ by $d_{G}(x)-p(x)$ for each vertex $x$ (including $z_{0}$). Then the resulting graph with the modified mapping satisfies $V(G)-z_{0}=V^{+}$ and is also a minimum member of $\mathcal{M}$.

For each vertex $x\in V(G)-z_{0}$,

[TABLE]

Claim 6. $d_{G}(z_{0})=k+p(z_{0})$, and all edges incident with $z_{0}$ are directed away from $z_{0}$.

By Claim 3, $z_{0}$ has a neighbour $x$. By Claim 5, $0<\alpha(x)<k/2$. If $xz_{0}$ is directed toward $z_{0}$, then we delete $xz_{0}$. By a proof similar to that of Claim 4, the resulting graph with modified mapping $p^{\prime}=p-\chi_{x}$ satisfies the conditions of the theorem. Since $|V(G^{\prime})|=|V(G)|$ and $|E(G^{\prime})|<|E(G)|$, and $(G,p,z_{0})$ is a smallest member of $\mathcal{M}$, the pre-orientation can be extended to a $p^{\prime}$-orientation of $G^{\prime}$ and then to a $p$-orientation of $G$ which contradicts the fact that $(G,p,z_{0})$ is a counterexample. So all edges incident with $z_{0}$ are directed away from $z_{0}$, and $d_{G}(z_{0})=d^{+}_{G}(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}p(z_{0})$. Now, we can assume that $d^{+}_{G}(z_{0})=d_{G}(z_{0})=sk+p(z_{0})$, where $s\geq 0$. By condition (i), we have $d_{G}(z_{0})\leq(2k-2)+2|\alpha(z_{0})|\leq 3k-2$ and so $s\leq 2$. In the case $s=2$, we derive that $p(z_{0})/2<|\alpha(z_{0})|$. Since

[TABLE]

we also derive that $\alpha(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}p(z_{0})/2$ which is a contradiction. By Claim 3, we have $d_{G}(z_{0})=k+p(z_{0})$. $\square$

The final step: $(G,p,z_{0})$ is not a counterexample.

By Claim 6, let $x$ be a neighbour of $z_{0}$, and let $e$ be an edge directed from $z_{0}$ to $x$. We replace $e$ by $k-1$ multiple directed edges from $x$ to $z_{0}$. Let $G^{\prime}$ be the resulting graph with $p^{\prime}=p-\chi_{x}-\chi_{z_{0}}$. We are going to prove that $G^{\prime}$ with the mapping $p^{\prime}$ satisfies all conditions of the theorem and, furthermore, $-k/2<\alpha^{\prime}(x)<0$ for the vertex $x$. By Claim 6, $d_{G}(z_{0})=k+p(z_{0})$. Since $p^{\prime}(z_{0})=p(z_{0})-1$ and $d_{G^{\prime}}(z_{0})=d_{G}(z_{0})+k-2$, we have

[TABLE]

This implies that $2|\alpha^{\prime}(z_{0})|=p(z_{0})$ and therefore, $d_{G^{\prime}}(z_{0})=(2k-2)+2|\alpha^{\prime}(z_{0})|$. So, condition (i) is satisfied for $G^{\prime}$ and $p^{\prime}$.

For condition (ii), we only need to consider $x$ and vertex sets containing $x$. Since $|\alpha(x)|\geq 1/2$, we have

[TABLE]

and hence $d_{G^{\prime}}(x)\geq(2k-2)+2|\alpha^{\prime}(x)|$. In addition,

[TABLE]

Since $0<\alpha(x)<k/2$, we have $\alpha^{\prime}(x)=\alpha(x)-k/2$ and so $-k/2<\alpha^{\prime}(x)<0$. By Claim 1, for any non-trivial vertex set $A$ of $G$ described in condition (ii) and containing $x$, we also have

[TABLE]

So, condition (ii) is also satisfied.

Now if $(G^{\prime},p^{\prime},z_{0})$ is also a counterexample, then $(G^{\prime},p^{\prime},z_{0})\in\mathcal{M}$, since $|V(G^{\prime})|=|V(G)|$ and $|E(G^{\prime}-z_{0})|=|E(G-z_{0})|$. But we have $V^{\prime+}=V(G^{\prime})-\{z_{0},x\}$ and $V^{\prime-}=\{x\}$, a contradiction to Claim 5. So $(G^{\prime},p^{\prime},z_{0})$ is not a counterexample, and hence $G^{\prime}$ has a $p^{\prime}$-orientation. Then the corresponding orientation of $G$ (obtained by replacing the $k-1$ edges from $x$ to $z_{0}$ with one edge in opposite direction) is a $p$-orientation of $G$ satisfying the theorem. This completes the proof.

• $\Box$*

When $z_{0}$ does not have small enough degree, one can replace the following version of Theorem 3.3. Note that by ignoring the extra condition on $z_{0}$, the proof can easily be obtained after adding an additional vertex of degree zero which plays the role of the vertex $z_{0}$ in Theorem 3.3.

Corollary 3.4

.* Let $G$ be a graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If for every vertex set $A$ with $\emptyset\subsetneq A\subsetneq V(G)$, $d_{G}(A)\geq 2k-2+2|\alpha(A)|,$ then $G$ has a $p$-orientation such that for each vertex $v$,*

[TABLE]

Furthermore, for an arbitrary vertex $z$, $d^{+}_{G}(z)$ can be assigned to any plausible integer value in whose interval.

**Proof. **

The proof is by induction on $|V(G)|+|E(G)|$. For $|V(G)|\leq 2$, the proof is trivial. Hence we may assume that $|V(G)|\geq 3$. If $d_{G}(z)=2k-2+2|\alpha(z)|$, then the proof can easily be derived from Theorem 3.3. So, suppose $d_{G}(z)\geq 2k+2|\alpha(z)|$. We claim that for any vertex set $A\subsetneq V(G)\setminus\{z\}$ with $|A|\geq 2$, we have $d_{G}(A)\geq 2k+2|\alpha(A)|$ and so $d_{G}(A^{c})\geq 2k+2|\alpha(A^{c})|$. For, if $d_{G}(A)<2k+2|\alpha(A)|$, then by a proof similar to that of Claim 1, we apply induction to $G/A$ and then we apply Theorem 3.3 to $G/A^{c}$. If $z$ has at least two neighbours, then we lift one non-parallel pair of edges incident with $z$. Otherwise, if $z$ has only one neighbour $y$, we lift one parallel pair of edges incident with $z$ and $y$. In this case, we have

[TABLE]

By applying the induction hypothesis the proof can be completed.

• $\Box$*

Corollary 3.5

.* Let $G$ be a graph, let $k$ be a positive integer, let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(3k-3)$-edge-connected, then it has a $p$-orientation such that for each vertex $v$,*

[TABLE]

Furthermore, for an arbitrary vertex $z$, $d^{+}_{G}(z)$ can be assigned to any plausible integer value in whose interval.

**Proof. **

The proof of $k=1$ is clear; note that the extra condition on $z$ can be obtained by reversing the orientation (if necessary). The proof of $k=2$ follows from Theorem 2.3. The proof of $k=3$ follows from Corollary 3.4. Note that the condition $|d^{+}_{G}(v)-d_{G}(v)/2|\leq k-1+|\alpha(v)|$ directly implies that $|d^{+}_{G}(v)-d_{G}(v)/2|<k$. Equivalently, $\lfloor\frac{d_{G}(v)}{2}\rfloor-(k-1)\leq d^{+}_{G}(v)\leq\lceil\frac{d_{G}(v)}{2}\rceil+(k-1)$, because $d^{+}_{G}(v)-d_{G}(v)/2\in\{\pm|\alpha(v)|,\pm(k-|\alpha(v)|)\}$ and also $d^{+}_{G}(v)-d_{G}(v)/2=0$ when $\alpha(v)=0$.

• $\Box$*

3.3 Replacing odd-edge-connectivity condition

Motivated by Theorem 4.12 in [9], we improve Theorem 3.3 as the following strengthened version which discounts the condition $d_{G}(A)\geq 2k-2+2|\alpha(A)|$ for any vertex set $A$ with $\alpha(A)=0$.

Theorem 3.6

.* Let $G$ be a graph with $z_{0}\in V(G)$, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Let $D_{z_{0}}$ be a pre-orientation of $E(z_{0})$ that is the set of edges incident with $z_{0}$. Assume that *

1. $(i)$

$\alpha(z_{0})\neq 0$. 2. $(ii)$

$d_{G}(z_{0})\leq 2k-2+2|\alpha(z_{0})|$, and the edges incident with $z_{0}$ are pre-directed such that $d^{+}_{G}(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}p(z_{0})$. 3. $(iii)$

$d_{G}(A)\geq 2k-2+2|\alpha(A)|$, for any vertex set $A$ with $\emptyset\subsetneq A\subsetneq V(G)\setminus\{z_{0}\}$ and $\alpha(A)\neq 0$.

Then the pre-orientation $D_{z_{0}}$ can be extended to a $p$-orientation $D$ of $G$ such that for each vertex $v$,

[TABLE]

**Proof. **

The proof is by induction on $|V(G)|+|E(G)|$. For $|V(G)|\leq 3$, the assertion holds by Theorem 3.3. So, suppose $|V(G)|\geq 4$. We claim that for any vertex set $A\subsetneq V(G)\setminus\{z_{0}\}$ such that $\alpha(A)\neq 0$ and $|A|\geq 2$, we have $d_{G}(A)\geq 2k+2|\alpha(A)|$. For, if $d_{G}(A)<2k+2|\alpha(A)|$, then by a proof similar to that of Claim 1, we apply induction to $G/A$ and then to $G/A^{c}$. Then by a proof similar to that of Claim 2, we claim that there is no vertex $v_{0}$ of $G$ such that $\alpha(v_{0})=0$; for otherwise we either remove $v_{0}$ or lift one pair of edges incident with $v_{0}$, and next we apply induction. Now $G$ must have a vertex set $A\subsetneq V(G)\setminus\{z_{0}\}$ such that $\alpha(A)=0$, $|A|\geq 2$, and $d_{G}(A)\leq 2k-2$. For otherwise $G$ satisfies the conditions of Theorem 3.3, and Theorem 3.6 follows. Choose $A$ with minimal $|A|$. We contract $A$ and use induction. Then we contract $A^{c}$ and by the minimality of $A$ we can apply Theorem 3.3 to the graph $G/A^{c}$.

• $\Box$*

When $z_{0}$ does not have small enough degree, one can replace the following version of Theorem 3.6.

Corollary 3.7

.* Let $G$ be a graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If for every vertex set $A$ with $\alpha(A)\neq 0$, $d_{G}(A)\geq 2k-2+2|\alpha(A)|,$ then $G$ has a $p$-orientation such that for each vertex $v$,*

[TABLE]

Furthermore, for an arbitrary vertex $z$, $d^{+}_{G}(z)$ can be assigned to any plausible integer value in whose interval.

**Proof. **

By induction on $|V(G)|+|E(G)|$. For $|V(G)|\leq 2$, the proof is trivial. Hence we may assume that $|V(G)|\geq 3$. Note that if $\alpha=0$, then the graph $G$ whose degrees are even and the theorem clearly holds. Also if $\alpha(z)=0$, then we can take another vertex as $z$ without this property. Hence we may assume that $\alpha(z)\neq 0$. If $d_{G}(z)=2k-2+2|\alpha(z)|$, then the conclusion trivially holds, using Theorem 3.6. So, suppose $d_{G}(z)\geq 2k+2|\alpha(z)|$. We claim that for any vertex set $A\subsetneq V(G)\setminus\{z\}$ such that $\alpha(A)\neq 0$ and $|A|\geq 2$, we have $d_{G}(A)\geq 2k+2|\alpha(A)|$ and so $d_{G}(A^{c})\geq 2k+2|\alpha(A^{c})|$. For, if $d_{G}(A)<2k+2|\alpha(A)|$, then by a proof similar to that of Claim 1, we apply induction to $G/A$ and then we apply Theorem 3.6 to $G/A^{c}$. If $z$ has at least two neighbours, then we lift one non-parallel pair of edges incident with $z$. Otherwise, if $z$ has only one neighbour $y$, we lift one parallel pair of edges incident with $z$ and $y$. In this case, we have

[TABLE]

By applying the induction hypothesis the proof can be completed.

• $\Box$*

Corollary 3.8

.* Let $G$ be a graph and let $k$ be an odd positive integer. If $G$ is odd-$(3k-2)$-edge-connected, then it has an orientation such that for each vertex $v$,*

[TABLE]

**Proof. **

The proof of $k=1$ is clear. So, suppose $k\geq 3$. For each vertex $v$ with even degree, define $p=0$ (mod $k$), and define $p=d_{G}(v)/2+k/2$ (mod $k$) otherwise. By Corollary 3.7, the graph $G$ has a $p$-orientation such that for each vertex $v$, $|d^{+}_{G}(v)-d_{G}(v)/2|<k$. This can complete the proof.

• $\Box$*

Corollary 3.9

.* Let $G$ be a graph with even degrees, let $k$ be a positive integer, and let $Q\subseteq V(G)$ with $|Q|$ even. If for every $A\subseteq V(G)$ with $|A\cap Q|$ odd, $d_{G}(A)\geq 6k-2,$ then $G$ has an orientation such that for each vertex $v$,*

[TABLE]

**Proof. **

For each $v\in Q$, define $p=d_{G}(v)/2+k$ (mod $2k$), and for each $v\in V(G)\setminus Q$, define $p=d_{G}(v)/2$ (mod $2k$). It is easy to check that for every vertex set $A$, we have $\alpha_{G}(A,p)=0$ when $|A\cap Q|$ is even, and $|\alpha_{G}(A,p)|=k$ when $|A\cap Q|$ is odd. Thus by Corollary 3.7, the graph $G$ has a $p$-orientation modulo $2k$. For the special case $k=1$, we can replace the condition $d_{G}(A)\geq 2$ for every vertex set $A$ with $|A\cap Q|$ odd. For this purpose, we need to apply Theorem 2.3 to each component of $G$ separately.

• $\Box$*

3.4 A new presentation: a lower bound independent of degrees of vertices

Our aim in this subsection is to introduce the following equivalent version of Theorem 3.3 which is useful for working with essentially edge-connected graphs. This version can directly be proved using the same arguments stated in the proof of Theorem 3.3. However, some parts in the proof would be shorter to state, some parts need more extra efforts.

Theorem 3.10

.* Let $G$ be a graph with $z_{0}\in V(G)$, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Let $D_{z_{0}}$ be a pre-orientation of $E(z_{0})$ that is the set of edges incident with $z_{0}$. Let $V_{0}=\{v\in V(G)-z_{0}:p(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}(v)/2\}$. If $V_{0}\neq\emptyset$, we let $v_{0}$ be a vertex of $V_{0}$ with smallest degree. Assume that*

1. $(i)$

$d_{G}(z_{0})\leq 2k-1+p(z_{0})$, and the edges incident with $z_{0}$ are pre-directed such that $d^{+}_{G}(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}p(z_{0})$. 2. $(ii)$

$d_{G}(A)\geq 2k-1+[p(A)]_{k}$, for any vertex set $A$ with $\emptyset\subsetneq A\subsetneq V(G)\setminus\{z_{0}\}$ and $A\neq\{v_{0}\}$.

Then the pre-orientation $D_{z_{0}}$ can be extended to a $p$-orientation $D$ of $G$ such that for each vertex $v$,

[TABLE]

Corollary 3.11

.* Let $G$ be a graph, let $k$ be an integer, $k\geq 3$, let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is essentially $(3k-3)$-edge-connected and for each vertex $v$, $d_{G}(v)\geq 2k-1+[p(v)]_{k}$, then $G$ has a $p$-orientation such that for each vertex $v$,*

[TABLE]

Furthermore, for an arbitrary vertex $z$, $d^{+}_{G}(z)$ can be assigned to any plausible integer value in whose interval.

Corollary 3.12

.* Let $G$ be a graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$ such that for each vertex $v$, $p(v)=0$ or $p(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}(v)$. If $G$ is $(2k-1)$-edge-connected essentially $(3k-3)$-edge-connected, then it has a $p$-orientation.*

The following proposition gives some useful relationship between the set functions $\alpha(A)$ and $[p(A)]_{k}$, which is a useful tool to show that why Theorems 3.3 and 3.10 are equivalent.

Proposition 3.13

.* Let $k$ be a positive integer and let $d$ and $p$ be two nonnegative integers with $p\in Z_{k}$. Take $\alpha$ to be a rational number such that $\alpha\in\{0,\pm 1/2,\ldots,\pm k/2\}$ and $\alpha\stackrel{{\scriptstyle k}}{{\equiv}}p-d/2$. Assume that $2k-2\leq d\leq 3k-2$. Then following statements hold:*

$d\leq 2k-2+2|\alpha|$* if and only if $d\leq 2k-1+p$.* 2. 2.

$d=2k-2+2|\alpha|$* if and only if $d=2k-1+p$ or $p=k-1$.* 3. 3.

$d\geq 2k-2+2|\alpha|$* if and only if $d\geq 2k-1+[p]_{k}$ .* 4. 4.

$d\geq 2k-1+[p]_{k}$* if and only if $d\geq 2k-1+[d-p]_{k}$.*

**Proof. **

Let $a$ be the unique integer in $\{-1,\ldots,k-1\}$ such that $d=2k-1+a$. It is not difficult to show that

[TABLE]

This can confirm items (1), (2), and (3). Now, assume that $d\geq 2k-1+[p]_{k}$ so that $a\geq[p]_{k}$. Thus $0\leq p\leq a$ or $p=k-1$ which implies that $a\geq[a-1-p]_{k}=[d-p]_{k}$. This can complete the proof.

• $\Box$*

4 Orientations modulo $k$: partition-connected versions

In this section, we improve the needed edge-connectivity in Corollary 3.5, but require the graph to have many edge-disjoint spanning trees.

4.1 Basic tools: Lifting operations preserving partition-connectivity

In this subsection, we present a sufficient condition for the existence of lifting operations which preserves tree-connectivity.

Theorem 4.1

.* Let $G$ be a general graph with $z\in V(G)$ and let $l_{0}$ be a nonnegative integer-valued function on $V(G)$. Assume that $z$ is not incident with loops. If $G$ contains an $(m,l_{0})$-partition-connected factor $H$ with $d_{G}(z)\geq 2d_{H}(z)-2l_{0}(z)-2m$, then there are $d_{H}(z)-l_{0}(z)-m$ pair of edges incident with $z$ such that by lifting them the resulting general graph $G_{0}$ with $V(G_{0})=V(G)\setminus\{z\}$ is still $(m,l_{0})$-partition-connected.*

**Proof. **

First assume that $m=0$. By the assumption, the graph $H$ has an orientation such for each vertex $v$, $d^{+}_{H}(v)\geq l_{0}(v)$. Let $u_{1}z,\ldots,u_{n}z$ be the edges of $H$ incident with $z$ directed toward $z$. Since $d_{G}(z)\geq 2(d_{H}(z)-l_{0}(z))\geq 2d^{-}_{H}(z)$, there are at least $n$ edges $v_{1}z,\ldots,v_{n}z$ of $E(G)\setminus\{u_{1}z,\ldots,u_{n}z\}$ incident with $z$. Define $G_{0}$ to be the directed general graph with $V(G_{0})=V(G)-z$ consisting of all directed edges of $G-z$ along with the new directed edges $u_{1}v_{1},\ldots,v_{n}u_{n}$ which $u_{i}v_{i}$ is directed from $u_{i}$ to $v_{i}$. It is easy to check that $G_{0}$ is $(0,l_{0})$-partition-connected.

Now, assume that $m=1$ and $l_{0}=0$. We prove this case by induction on the number of components of $H-z$. Let $C_{1},\ldots,C_{n}$ be the components of $H-z$ and let $u_{1}z,\ldots,u_{n}z$ be $n$ edges of $H$ incident with $z$ such that $u_{i}\in V(C_{i})$. If $n=1$, then $G-z$ is connected and the proof is clear. If $n=2$, define $G_{0}$ to be the graph with $V(G_{0})=V(G)-z$ consisting of all edges of $G-z$ along with the new edge $u_{1}u_{2}$. It is easy to check that $G_{0}$ is connected. So, suppose $n\geq 3$. Since $d_{G}(z)>d_{H}(z)$, there is an edge $v_{1}z$ of $E(G)\setminus E(H)$ incident with $z$. We may assume that $v_{1}\not\in V(C_{1})$. Define $G^{\prime}$ and $H^{\prime}$ to be the graphs obtained from $G$ and $H$ by removing the edges $u_{1}z$ and $v_{1}z$ and adding the new edge $u_{1}v_{1}$. It is easy to check that $H^{\prime}$ is still connected and $H^{\prime}-z$ has $n-1$ components. Moreover, we have $d_{G^{\prime}}(z)=d_{G}(z)-2$ and $d_{H^{\prime}}(z)=d_{H}(z)-1$, which implies that $d_{G^{\prime}}(z)\geq 2d_{H^{\prime}}(z)-2$. Now, by applying induction to the graphs $G^{\prime}$ and $H^{\prime}$ the proof of this part can easily be completed.

Now, we are going to prove the remaining cases by induction on $m$. Suppose $m\geq 1$. Set $m_{1}=1$, $l_{1}=0$, $m_{2}=m-1$, and $l_{2}=l_{0}$. Decompose $H$ into two factors $H_{1}$ and $H_{2}$ such that each $H_{i}$ is $(l_{i}+m_{i})$-partition-connected. Also, decompose $G$ into two factors $G_{1}$ and $G_{2}$ such that each $G_{i}$ contains $H_{i}$. Since $d_{G}(z)\geq 2d_{H}(z)-2l_{0}(z)-2m$, we must have $d_{G_{i}}(z)\geq 2d_{H_{i}}(z)-2l_{i}(z)-2m_{i}$ for at least an integer $i\in\{1,2\}$. Let $j\in\{1,2\}$ with $j\neq i$. By induction hypothesis, there are $d_{H_{i}}(z)-l_{i}(z)-m_{i}$ pair of edges of $G_{i}$ incident with $z$ such that by lifting them the resulting general graph $G^{\prime}_{i}$ with $V(G^{\prime}_{i})=V(G_{i})\setminus\{z\}$ is still $(m_{i},l_{i})$-partition-connected. Define $\mathcal{G}_{j}$ to be the factor of $G$ consisting of all edges of $G_{j}$ together with all remaining edges of $G_{i}$ incident with $z$ which are not lifted. According to the construction of $\mathcal{G}_{j}$, we must have $d_{\mathcal{G}_{j}}(z)=d_{G}(z)-2(d_{H_{i}}(z)-l_{i}(z)-m_{i})\geq 2d_{H_{j}}(z)-2l_{j}(z)-2m_{j}$. By induction hypothesis, there are $d_{H_{j}}(z)-l_{j}(z)-m_{j}$ pair of edges of $\mathcal{G}_{j}$ incident with $z$ such that by lifting them the resulting general graph $G^{\prime}_{j}$ with $V(G^{\prime}_{j})=V(\mathcal{G}_{j})\setminus\{z\}$ is still $(m_{j},l_{j})$-partition-connected. It is enough, now, to define $G_{0}=G^{\prime}_{1}\cup G^{\prime}_{2}$. Hence the theorem is proved.

• $\Box$*

Corollary 4.2

.([3])* Let $G$ be an $m$-tree-connected graph with $z\in V(G)$. If $d_{G}(z)\leq 2m$, then there are $d_{G}(z)-m$ non-parallel pair of edges incident with $z$ such that by lifting them the resulting graph $G_{0}$ with $V(G_{0})=V(G)\setminus\{z\}$ is still $m$-tree-connected.*

4.2 Graphs with partition-connectivity at least $(2k-2,l_{0})$

In this subsection, we improve the needed edge-connectivity in Corollary 3.5, but require the graph to have many edge-disjoint spanning trees.

Theorem 4.3

.* Let $G$ be a general graph with $z\in V(G)$, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Let $s$, $s_{0}$, and $l_{0}$ be three integer-valued functions on $V(G)$ satisfying $s+s_{0}+k-1\leq d_{G}$ and $\max\{s,s_{0}\}\leq l_{0}+(k-1)(1-\chi_{z})$. If $G$ is $(2k-2,l_{0})$-partition-connected, then it has a $p$-orientation such that for each vertex $v$,*

[TABLE]

**Proof. **

We may assume that $l_{0}$ is nonnegative and $G$ is loopless. The proof is by induction on $|V(G)|$. For $|V(G)|\leq 2$ the proof is straightforward. So, suppose $|V(G)|\geq 3$. For notational simplicity, let us define $m=2k-2$. For proving the theorem, we shall consider the following four cases.

Cases 1. There is a vertex $u\in V(G)\setminus\{z\}$ with $d_{G}(u)=2l_{0}(u)+2m-r$ such that $0\leq r\leq l_{0}(u)+m$ and $p(u)\stackrel{{\scriptstyle k}}{{\equiv}}l_{0}(u)+m-i$ where $0\leq i\leq\min\{r,k-1\}$.

By Theorem 4.1, there are $l_{0}(u)+m-r$ pair of edges incident with $u$ such that by lifting them the resulting general graph $H$ with $V(H)=V(G)\setminus\{u\}$ is still $(m,l_{0})$-partition-connected. Obviously, $d_{R}(u)=d_{G}(u)-2(l_{0}(u)+m-r)=r$, where $R$ is the factor of $G$ consisting of all edges incident with $u$ that are not lifted. Since $i\leq r$, the edges of $R$ can be orientated such that $d^{+}_{R}(u)\stackrel{{\scriptstyle k}}{{\equiv}}r-i$. Define $s^{\prime}(u)=s(u)-(l_{0}(u)+m-r)$ and $s^{\prime}_{0}(u)=s_{0}(u)-(l_{0}(u)+m-r)$. By the assumption, we must have $\max\{s^{\prime}(u),s^{\prime}_{0}(u)\}\leq d_{R}(u)-(k-1)$, and $s^{\prime}(u)+s^{\prime}_{0}(u)\leq d_{R}(u)-(k-1)$. Therefore, if $d_{R}(u)\geq k-1$ then the orientation of $R$ can be selected such that $s^{\prime}(u)\leq d^{+}_{R}(u)\leq d_{R}(u)-s^{\prime}_{0}(u)$. If $d_{R}(u)\leq k-1$, then $\max\{s(v),s_{0}(v)\}\leq 0$ and so we must automatically have

[TABLE]

Now, for each vertex $v$ of $H$, define $p^{\prime}(v)=p(v)-d^{+}_{R}(v)$. It is easy to check that $|E(H)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(H)}p^{\prime}(v)$. Obviously, $\max\{s(v)-d_{R}^{+}(v),s_{0}(v)-d_{R}^{-}(v)\}\leq\max\{s(v),s_{0}(v)\}\leq l_{0}(v)+k-1$ and $s(v)-d_{R}^{+}(v)+s_{0}(v)-d_{R}^{-}(v)+k-1\leq d_{H}(v)$. Thus by the induction hypothesis, $H$ has a $p^{\prime}$-orientation modulo $k$ such that for each $v\in V(H)$,

[TABLE]

This orientation induces a $p$-orientation for $G$ such that for each $v\in V(H)$, $d^{+}_{G}(v)=d^{+}_{H}(v)+d^{+}_{R}(v)$, and also $d_{G}^{+}(u)=d^{+}_{R}(u)+l_{0}(u)+m-r.$ This can complete the proof of Case 1. $\square$

Case 2. $d_{G}(z)<2l_{0}(z)+k-1$.

Since $d_{G}(z)\geq 2k-2$, we must have $l_{0}(z)>0$ and hence there is an edge $zu$ incident with $z$ such that the graph $G_{0}$ is $(m,l_{0}-\chi_{z})$-partition-connected, where $G_{0}=G-zu$.

First assume that $s(z)<l_{0}(z)$ which implies that $s(z)\leq l_{0}(z)-\chi_{z}(z)$. Thus by the induction hypothesis, the graph $G_{0}$ has a $(p-\chi_{u})$-orientation such that for each vertex $v$, $s(v)-\chi_{u}(v)\leq d_{G_{0}}^{+}(v)\leq d_{G_{0}}(v)-(s_{0}(v)-\chi_{z}(v))$. Now, this orientation induces the desired $p$-orientation for $G$ by adding an edge directed from $u$ to $z$.

Now, assume that $s(z)=l_{0}(z)$. This implies that $s_{0}(z)<l_{0}(z)$, because $s(z)+s_{0}(z)+k-1\leq d_{G}(z)<2l_{0}(z)+k-1$. Thus by the induction hypothesis, the graph $G_{0}$ has a $(p-\chi_{z})$-orientation such that for each vertex $v$, $s(v)-\chi_{z}(v)\leq d_{G_{0}}^{+}(v)\leq d_{G_{0}}(v)-(s_{0}(v)-\chi_{u}(v))$. Now, this orientation induces the desired $p$-orientation for $G$ by adding an edge directed from $z$ to $u$. This completes the proof of Case 2. $\square$

Case 3. There is a nonempty proper subset $A$ of $V(G)\setminus\{z\}$ such that $d_{G}(A)<2k-2+2|\alpha(A)|$.

By the first case, we must have $|A|\geq 2$, because $d_{G}(v)\geq 2l_{0}(v)+3k-3$ for each $v\in V(G)\setminus\{z\}$. Choose $A$ with minimal $|A|$. We contract $A$ and use induction. Note that $G/A$ is also $(m,l_{0})$-partition-connected, where $l_{0}(A)=0$. Then we contract $A^{c}$ and by the minimality of $A$, we can apply Theorem 3.3 to the graph $G/A^{c}$. For verifying the condition on out-degrees of vertices of $A$, we can apply the same arguments stated in the next case. $\square$

Case 4: For every nonempty proper subset $A$ of $V(G)\setminus\{z\}$, $d_{G}(A)\geq 2k-2+2|\alpha(A)|$.

By applying Theorem 3.3 or Corollary 3.4 (with respect to the case that $d_{G}(z)\leq 2k-2+2|\alpha(z)|$ or not), the graph $G$ has a $p$-orientation such that $|d^{+}_{G}(z)-d_{G}(z)/2|\leq k/2$ and $|d^{+}_{G}(v)-d_{G}(v)/2|<k$ for all vertices $v$. According to Case 2, $d_{G}(z)\geq 2l_{0}(z)+k-1$ which implies that $s_{0}(z)\leq l_{0}(z)\leq d^{+}_{G}(z)\leq d_{G}(z)-l_{0}(z)\leq d_{G}(z)-s_{0}(z)$. Let $v\in V(G)\setminus\{z\}$. If $d_{G}(v)\geq 2l_{0}(v)+2m$, then we must automatically have $s(v)\leq l_{0}(v)+k-1\leq d^{+}_{G}(v)\leq d_{G}(v)-l_{0}(v)-(k-1)\leq d_{G}(v)-s_{0}(v).$ Otherwise, $d_{G}(v)=2l_{0}(v)+2m-r$ in which $0<r<k-1$ and $l_{0}(v)+m-k-r/2<d^{+}_{G}(v)<l_{0}(v)+m+k-r/2$. According to Case 1,

[TABLE]

which again implies that $s(v)\leq l_{0}(v)+m-(k-1)\leq d^{+}_{G}(v)\leq d_{G}(v)-l_{0}(v)-m+(k-1)\leq d_{G}(v)-s_{0}(v).$ Hence the proof is completed.

• $\Box$*

In the following theorem, we shall restate a simpler version of Theorem 4.3 which is refined by involving extension of pre-orientations.

Theorem 4.4

.* Let $G$ be a non-trivial graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(2k-2)$-tree-connected, then it has a $p$-orientation such that for each vertex $v$,*

[TABLE]

**Proof. **

The proof can be obtained by induction on $|V(G)|$ as the arguments stated in the proof Theorem 4.3. For proving the assertion, one can consider the following three cases: Case (i) there is a vertex $u$ with $d_{G}(u)\leq 3k-3$. In this case, we should apply Theorem 4.2. Case (ii) there is a vertex set $A$ satisfying $2\leq|A|\leq|V(G)|-2$ and $d_{G}(A)\leq 2k-2+2|\alpha(A)|$. In this case, we choose $A$ with minimal $|A|$. We contract $A$ and use induction. Note that $G/A$ is also $(2k-2)$-tree-connected. Then we contract $A^{c}$ and by the minimality of $A$, we can apply Theorem 3.3 to the graph $G/A^{c}$. Case (iii) fore every nonempty proper subset $A$ of $V(G)$, $d_{G}(A)\geq 2k-2+2|\alpha(A)|$. In this case, we should apply Corollary 3.4.

• $\Box$*

Corollary 4.5

.* Let $G$ be a non-trivial graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $(2k-2)$-tree-connected, then it has a $p$-orientation such that for each vertex $v$, $0<d^{+}_{G}(v)<d_{G}(v).$*

**Proof. **

Apply Theorem 4.4.

• $\Box$*

4.3 A generalization: definition of $\lambda_{k}$

In this subsection, we shall define a parameter $\lambda_{k}$ and give an application of it on tree-connected graphs. For this purpose, we need to form the following consequence of Theorem 3.3 to show that the definition is well-defined by giving an upper bound on it. The following fact can be considered as an extension of Lemma 2.2 (i) in [6].

Theorem 4.6

.* Let $G$ be a graph with $z_{0}\in V(G)$, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. Let $D$ be a given pre-orientation of the edges incident with $z_{0}$ such that $d^{+}_{D}(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}p(z_{0})$. If $d_{G}(z_{0})\leq 3k-2$ and $d_{G}(A)\geq 3k-3$ for every nonempty proper subset $A$ of $V(G)\setminus\{z_{0}\}$, then pre-orientation of $D$ can be extended a $p$-orientation of $G$.*

**Proof. **

Let $D^{\prime}$ be another pre-orientation of the edges incident with $z_{0}$ such that $d^{+}_{D^{\prime}}(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}k-1$. For each vertex $v$, we define $p^{\prime}(v)=p(v)-d^{+}_{D}(v)+d^{+}_{D^{\prime}}(v)$ (mod $k$). Since $\sum_{v\in V(G)}d^{+}_{D}(v)\stackrel{{\scriptstyle k}}{{\equiv}}d_{G}(z_{0})\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}d^{+}_{D^{\prime}}(v)$, it is easy to check that $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p^{\prime}(v)$. Since $d_{G}(z_{0})\leq 3k-2=2k-1+p^{\prime}(z_{0})$, by Theorem 3.10, the pre-orientation of $D^{\prime}$ can be extended to a $p^{\prime}$-orientation modulo $k$ of $G$. Now, it is enough to replace the orientation of the edges of $D$ in the current orientation of $G$ to obtain the desired $p$-orientation.

• $\Box$*

For any positive integer $k$ with $k\geq 3$, we define $\lambda_{k}$ to be the smallest positive integer such that the following holds: If $G$ is a graph with $z_{0}\in V(G)$ satisfying $d_{G}(z_{0})<\lambda_{k}$ and $d_{G}(A)\geq\lambda_{k}$ for every nonempty proper subset $A$ of $V(G)\setminus\{z_{0}\}$, then any suitable pre-orientation of the edges incident with $z_{0}$ can be extended a $p$-orientation, where $p:V(G)\rightarrow Z_{k}$ is a given arbitrary mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. According the definition of $\lambda_{k}$, one can now formulate the following result on modulo orientation of tree-connected graphs. Note that for defining $\lambda_{k}$, we could also restrict our attention to $\lceil(\lambda_{k}+k-2)/2\rceil$-tree-connected graphs to deduce the following result alternatively.

Theorem 4.7

.* Let $G$ be a graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $\lceil(\lambda_{k}+k-2)/2\rceil$-tree-connected, then it has a $p$-orientation.*

**Proof. **

Let $m=\lceil(\lambda_{k}+k-2)/2\rceil$. For proving the theorem, we consider the following three cases: Case (i) there is a vertex $u$ with $d_{G}(u)<\lambda_{k}$. In this case, by Corollary 4.2, there are $d_{G}(u)-m$ non-parallel pair of edges incident with $u$ such that by lifting them the resulting graph $G_{0}$ with $V(G_{0})=V(G)\setminus\{u\}$ is still $m$-tree-connected. Thus $2m-d_{G}(u)$ edges are not lifted. Since $d_{G}(u)\leq\lambda_{k}-1$, we must have $2m-d_{G}(u)\geq 2m-\lambda_{k}+1\geq k-1$. Let $M$ be the factor of $G$ consisting of the edges incident with $u$ that are not lifted. Since $|E(M)|\geq k-1$, the edges of $M$ can be directed such that $d^{+}_{M}(u)+(d_{G}(u)-d_{M}(u))/2\stackrel{{\scriptstyle k}}{{\equiv}}p(u)$. Now, by applying the induction hypothesis on $G_{0}$, the orientation of $M$ can be extended to a $p$-orientation of $G$. Case (ii) there is a vertex set $A$ with $2\leq|A|\leq|V(G)|-2$ and $d_{G}(A)<\lambda_{k}$. Choose $A$ with minimal $|A|$. We contract $A$ and use induction. Then we contract $A^{c}$ and by minimality of $A$ and the definition of $\lambda_{k}$, the pre-orientation can be extended to a $p$-orientation of $G$. Note that $G/A$ is also $m$-tree-connected. Case (iii) $G$ is $\lambda_{k}$-edge-connected. This case also follows from the definition of $\lambda_{k}$ by adding an artificial vertex $z_{0}$.

• $\Box$*

In 2012 Barát, Gerbner, and Thomassé proposed a conjecture on star-decomposition of simple graphs which can be reformulated to the following modulo orientation version, see [8]. Recently, Han, Li, Wu, and Zhang [5] showed that the following conjecture cannot be developed to the class of $(k-1)$-tree-connected $(2k-1)$-edge-connected graphs, when $k\geq 11$. By the above-mentioned theorem, one can confirm this conjecture for $\lceil(3k-2)/2\rceil$-tree-connected graphs, if $\lambda_{k}\leq 2k$.

Conjecture 4.8

.([1])* Let $G$ be a graph, let $k$ be an integer, $k\geq 3$, and let $p:V(G)\rightarrow Z_{k}$ be a mapping with $|E(G)|\stackrel{{\scriptstyle k}}{{\equiv}}\sum_{v\in V(G)}p(v)$. If $G$ is $k$-tree-connected, then it has a $p$-orientation.*

Remark 4.9

. Recently, Esperet, de Joannis de Verclos, Le, and Thomassé (2018) [2] utilized Theorems 1.2 to establish a result on additive bases and a result on weighted orientations. By reviewing their proofs, we find out one can replace Theorem 4.3 in their proofs to get further improvements.