Additive bases and flows in graphs

TL;DR

This paper proves a conjecture about additive bases in finite vector spaces for vectors with at most two non-zero entries and applies this to establish new flow results in highly edge-connected graphs.

Contribution

It confirms the conjecture for vectors with limited non-zero entries and extends flow existence results in highly edge-connected graphs.

Findings

Confirmed the conjecture for vectors with at most two non-zero entries.

Established new flow existence results in highly edge-connected graphs.

Extended known results on flows in directed graphs with list constraints.

Abstract

It was conjectured by Jaeger, Linial, Payan, and Tarsi in 1992 that for any prime number $p$, there is a constant $c$ such that for any $n$, the union (with repetition) of the vectors of any family of $c$ linear bases of $\mathbb{Z}_p^n$ forms an additive basis of $\mathbb{Z}_p^n$ (i.e. any element of $\mathbb{Z}_p^n$ can be expressed as the sum of a subset of these vectors). In this note, we prove this conjecture when each vector contains at most two non-zero entries. As an application, we prove several results on flows in highly edge-connected graphs, extending known results. For instance, assume that $p\ge 3$ is a prime number and $\vec{G}$ is a directed, highly edge-connected graph in which each arc is given a list of two distinct values in $\mathbb{Z}_p$. Then $\vec{G}$ has a $\mathbb{Z}_p$-flow in which each arc is assigned a value of its own list.

Full text

The authors are partially supported by ANR Project STINT (anr-13-bs02-0007) and GATO (anr-16-ce40-0009-01), and LabEx PERSYVAL-Lab (anr-11-labx-0025). An extended abstract of this work appeared in the proceedings of Eurocomb 2017.

1. Introduction

Graphs considered in this paper may have multiple edges but no loops. An additive basis $B$ of a vector space $F$ is a multiset of elements from $F$ such that for all $\beta\in F$, there is a subset of $B$ which sums to $\beta$. Let $\mathbb{Z}_{p}^{n}$ be the $n$-dimensional linear space over the prime field $\mathbb{Z}_{p}$. The following result is a simple consequence of the Cauchy-Davenport Theorem [5] (see also [2]).

Theorem 1** ([5]).**

For any prime $p$, any multiset of $p-1$ non-zero elements of $\mathbb{Z}_{p}$ forms an additive basis of $\mathbb{Z}_{p}$.

This result can be rephrased as: for $n=1$, any family of $p-1$ linear bases of $\mathbb{Z}_{p}^{n}$ forms an additive basis of $\mathbb{Z}_{p}^{n}$. A natural question is whether this can be extended to all integers $n$. Given a collection of sets $X_{1},...,X_{k}$, we denote by $\biguplus_{i=1}^{k}X_{i}$ the union with repetitions of $X_{1},...,X_{k}$. Jaeger, Linial, Payan and Tarsi [12] conjectured the following, a generalization of important results regarding nowhere-zero flows in graphs.

Conjecture 2** ([12]).**

For every prime number $p$, there is a constant $c(p)$ such that for any $t\geqslant c(p)$ linear bases $B_{1},...,B_{t}$ of $\mathbb{Z}_{p}^{n}$, the union $\biguplus_{s=1}^{t}B_{s}$ forms an additive basis of $\mathbb{Z}_{p}^{n}$.

Alon, Linial and Meshulam [1] proved a weaker version of Conjecture 2, that the union of any $p\lceil\log n\rceil$ linear bases of $\mathbb{Z}_{p}^{n}$ contains an additive basis of $\mathbb{Z}_{p}^{n}$ (note that their bound depends on $n$). The support of a vector $x=(x_{1},\ldots,x_{n})\in\mathbb{Z}_{p}^{n}$ is the set of indices $i$ such that $x_{i}\neq 0$. The shadow of a vector $x$ is the (unordered) multiset of non-zero entries of $x$. Note that sizes of the support and of the shadow of a vector are equal. In this note, we prove that Conjecture 2 holds if the support of each vector has size at most two.

Theorem 3**.**

Let $p\geqslant 3$ be a prime number. For some integer $\ell\geqslant 1$, consider $t\geqslant 8\ell(3p-4)+p-2$ linear bases $B_{1},...,B_{t}$ of $\mathbb{Z}_{p}^{n}$, such that the support of each vector has size at most $2$, and at most $\ell$ different shadows of size $2$ appear among the vectors of $\mathcal{B}=\biguplus_{s=1}^{t}B_{s}$. Then $\mathcal{B}$ forms an additive basis of $\mathbb{Z}_{p}^{n}$.

Theorem 3 will be proved in Section 3 using a result of Lovász, Thomassen, Wu and Zhang [15] (Theorem 6 below) on flows in highly edge-connected graphs. It was mentioned to us by one of the referees that Lai and Li [14] established the equivalence between Theorem 6 and Theorem 3 in the special case where all the shadows are equal to $\{-1,+1\}\pmod{p}$.

The number of possibilities for an (unordered) multiset of $\mathbb{Z}_{p}\setminus\{0\}$ of size 2 is ${p-1\choose 2}+p-1={p\choose 2}$. As a consequence, Theorem 3 has the following immediate corollary.

Corollary 4**.**

Let $p\geqslant 3$ be a prime number. For any $t\geqslant 8{p\choose 2}(3p-4)+p-2$ linear bases $B_{1},...,B_{t}$ of $\mathbb{Z}_{p}^{n}$ such that the support of each vector has size at most 2, $\biguplus_{s=1}^{t}B_{s}$ forms an additive basis of $\mathbb{Z}_{p}^{n}$.

Another interesting consequence of Theorem 3 concerns the linear subspace $(\mathbb{Z}_{p}^{n})_{0}$ of vectors of $\mathbb{Z}_{p}^{n}$ whose entries sum to $0\pmod{p}$.

Corollary 5**.**

Let $p\geqslant 3$ be a prime number. For any $t\geqslant 4(p-1)(3p-4)+p-2$ linear bases $B_{1},...,B_{t}$ of $(\mathbb{Z}_{p}^{n})_{0}$ such that the support of each vector has size at most 2, $\biguplus_{s=1}^{t}B_{s}$ forms an additive basis of $(\mathbb{Z}_{p}^{n})_{0}$.

Proof.

Note that for any $1\leqslant s\leqslant t$, the linear basis $B_{s}$ consists of $n-1$ vectors, each of which has a support of size 2, and the two elements of the shadow sum to $0\pmod{p}$. In particular, at most $\tfrac{p-1}{2}$ different shadows appear among the vectors of the linear bases $B_{1},...,B_{t}$. It is convenient to view each $B_{s}$ as a matrix in which the elements of the basis are column vectors. For each $1\leqslant s\leqslant t$, let $B_{s}^{\prime}$ be obtained from $B_{s}$ by deleting the last row. It is easy to see that $B_{s}^{\prime}$ is a linear basis of $\mathbb{Z}^{n-1}_{p}$. Moreover, at most $\tfrac{p-1}{2}$ different shadows of size 2 appear among the vectors of the linear bases $B^{\prime}_{1},...,B^{\prime}_{t}$ (note that the removal of the last row may have created vectors with shadows of size 1). In particular, it follows from Theorem  3 that for any vector $\beta=(\beta_{1},\ldots,\beta_{n})\in(\mathbb{Z}_{p}^{n})_{0}$, the vector $(\beta_{1},\ldots,\beta_{n-1})\in\mathbb{Z}_{p}^{n-1}$ can be written as a sum of a subset of elements of $\biguplus_{s=1}^{t}B_{s}^{\prime}$. Clearly, the corresponding subset of elements of $\biguplus_{s=1}^{t}B_{s}$ sums to $\beta$. This concludes the proof of Corollary 5. ∎

In the next section, we explore some consequences of Corollary 5.

2. Orientations and flows in graphs

Let $G=(V,E)$ be a non-oriented graph. An orientation $\vec{G}=(V,\vec{E})$ of $G$ is obtained by giving each edge of $E$ a direction. For each edge $e\in E$, we denote the corresponding arc of $\vec{E}$ by $\vec{e}$, and vice versa. For a vertex $v\in V$, we denote by $\delta^{+}_{\vec{G}}(v)$ the set of arcs of $\vec{E}$ leaving $v$, and by $\delta^{-}_{\vec{G}}(v)$ the set of arcs of $\vec{E}$ entering $v$.

For an integer $k\geqslant 2$, a mapping $\beta:V\rightarrow\mathbb{Z}_{k}$ is said to be a $\mathbb{Z}_{k}$-boundary of $G$ if $\sum_{v\in V}\beta(v)\equiv 0\pmod{k}$. Given a $\mathbb{Z}_{k}$-boundary $\beta$ of $G$, an orientation $\vec{G}$ of $G$ is a $\beta$-orientation if $d_{\vec{G}}^{+}(v)-d_{\vec{G}}^{-}(v)\equiv\beta(v)\pmod{k}$ for every $v\in V$, where $d_{\vec{G}}^{+}(v)$ and $d_{\vec{G}}^{-}(v)$ stand for the out-degree and the in-degree of $v$ in $\vec{G}$.

The following major result was obtained by Lovász, Thomassen, Wu and Zhang [15]:

Theorem 6**.**

[15]** For any $k\geqslant 1$, any $6k$-edge-connected graph $G$, and any $\mathbb{Z}_{2k+1}$-boundary $\beta$ of $G$, the graph $G$ has a $\beta$-orientation.

A natural question is whether a weighted counterpart of Theorem 6 exists. Given a graph $G=(V,E)$, a $\mathbb{Z}_{k}$-boundary $\beta$ of $G$ and a mapping $f:E\rightarrow\mathbb{Z}_{k}$, an orientation $\vec{G}$ of $G$ is called an $f$-weighted $\beta$-orientation if $\partial f(v)\equiv\beta(v)\pmod{k}$ for every $v$, where $\partial f(v)=\sum_{\vec{e}\in\delta^{+}_{\vec{G}}(v)}f({e})-\sum_{\vec{e}\in\delta^{-}_{\vec{G}}(v)}f({e})$. Note that if $f(e)\equiv 1\pmod{k}$ for every edge $e$, an $f$-weighted $\beta$-orientation is precisely a $\beta$-orientation.

An immediate observation is that if we wish to have a general result of the form of Theorem 6 for weighted orientations, it is necessary to assume that $2k+1$ is a prime number. For instance, take $G$ to consist of two vertices $u,v$ with an arbitrary number of edges between $u$ and $v$, consider a non-trivial divisor $p$ of $2k+1$, and ask for a $\mathbf{p}$-weighted $\mathbb{Z}_{2k+1}$-orientation $\vec{G}$ of $G$ (here, $\mathbf{p}$ denotes the function that maps each edge to $p\pmod{2k+1}$). Note that for any orientation, $\partial{\mathbf{p}}(v)$ is in the subgroup of $\mathbb{Z}_{2k+1}$ generated by $p$, and this subgroup does not contain $1,-1\pmod{2k+1}$. In particular, there is no $\mathbf{p}$-weighted $\mathbb{Z}_{2k+1}$-orientation of $G$ with boundary $\beta$ satisfying $\beta(u)\equiv-\beta(v)\equiv 1\pmod{2k+1}$.

In Section 4, we will prove that Corollary 5 easily implies a weighted counterpart of Theorem 6 as in the following theorem, but with a stronger requirement on the edge-connectivity. Theorem 7 itself will be deduced directly from Theorem 6.

Theorem 7**.**

Let $p\geqslant 3$ be a prime number and let $G=(V,E)$ be a $(6p-8)(p-1)$-edge-connected graph. For any mapping $f:E\rightarrow\mathbb{Z}_{p}\setminus\{0\}$ and any $\mathbb{Z}_{p}$-boundary $\beta$, $G$ has an $f$-weighted $\beta$-orientation.

Theorem 7 turns out to be equivalent to the following seemingly more general result. Assume that we are given a directed graph $\vec{G}=(V,\vec{E})$ and a $\mathbb{Z}_{p}$-boundary $\beta$. A $\mathbb{Z}_{p}$-flow with boundary $\beta$ in $\vec{G}$ is a mapping $f:\vec{E}\rightarrow\mathbb{Z}_{p}$ such that $\partial f(v)\equiv\beta(v)\pmod{p}$ for every $v$. In other words, $f$ is a $\mathbb{Z}_{p}$-flow with boundary $\beta$ in $\vec{G}=(V,\vec{E})$ if and only if $\vec{G}$ is an $f$-weighted $\beta$-orientation of its underlying non-oriented graph $G=(V,E)$, where $f$ is extended from $\vec{E}$ to $E$ in the natural way (i.e. for each $e\in E$, $f(e):=f(\vec{e})$).

In the remainder of the paper we will say that a directed graph $\vec{G}$ is $t$-edge-connected if its underlying non-oriented graph, denoted by $G$, is $t$-edge-connected.

Theorem 8**.**

Let $p\geqslant 3$ be a prime number and let $\vec{G}=(V,\vec{E})$ be a directed $(6p-8)(p-1)$-edge-connected graph. For any arc $\vec{e}\in\vec{E}$, let $L(\vec{e})$ be a pair of distinct elements of $\mathbb{Z}_{p}$. Then for every $\mathbb{Z}_{p}$-boundary $\beta$, $\vec{G}$ has a $\mathbb{Z}_{p}$-flow $f$ with boundary $\beta$ such that for any $\vec{e}\in\vec{E}$, $f(\vec{e})\in L(\vec{e})$.

This result can been seen as a choosability version of Theorem 6 (the reader is referred to [6] for choosability versions of some classical results on flows). To see that Theorem 8 implies Theorem 7, simply fix an arbitrary orientation of $G$ and set $L(\vec{e})=\{f(e),-f(e)\}$ for each arc $\vec{e}$. We now prove that Theorem 7 implies Theorem 8. We actually prove a slightly stronger statement (holding in $\mathbb{Z}_{2k+1}$ for any integer $k\geqslant 1$).

Lemma 9**.**

Let $k\geqslant 1$ be an integer, and let $\vec{G}=(V,\vec{E})$ be a directed graph such that the underlying non-oriented graph $G$ has an $f$-weighted $\beta$-orientation for any mapping $f:E\rightarrow\mathbb{Z}_{2k+1}\setminus\{0\}$ and any $\mathbb{Z}_{2k+1}$-boundary $\beta$. For every arc $\vec{e}\in\vec{E}$, let $L(\vec{e})$ be a pair of distinct elements of $\mathbb{Z}_{2k+1}$. Then for every $\mathbb{Z}_{2k+1}$-boundary $\beta$, $\vec{G}$ has a $\mathbb{Z}_{2k+1}$-flow $g$ with boundary $\beta$ such that $g(\vec{e})\in L(\vec{e})$ for every $\vec{e}$.

Proof.

Let $\beta$ be a $\mathbb{Z}_{2k+1}$-boundary of $\vec{G}$. Consider a single arc $\vec{e}=(u,v)$ of $\vec{G}$. Choosing one of the two values of $L(\vec{e})$, say $a$ or $b$, will either add $a$ to $\partial g(u)$ and subtract $a$ from $\partial g(v)$, or add $b$ to $\partial g(u)$ and subtract $b$ from $\partial g(v)$. Note that 2 and $2k+1$ are relatively prime, so the element $2^{-1}$ is well-defined in $\mathbb{Z}_{2k+1}$. If we now add $2^{-1}(a+b)$ to $\beta(v)$ and subtract $2^{-1}(a+b)$ from $\beta(u)$, the earlier choice is equivalent to choosing between the two following options: adding $2^{-1}(a-b)$ to $\partial g(u)$ and subtracting $2^{-1}(a-b)$ from $\partial g(v)$, or adding $2^{-1}(b-a)$ to $\partial g(u)$ and subtracting $2^{-1}(b-a)$ from $\partial g(v)$. This is equivalent to choosing an orientation for an edge of weight $2^{-1}(a-b)$. It follows that finding a $\mathbb{Z}_{2k+1}$-flow $g$ with boundary $\beta$ such that for any $\vec{e}\in\vec{E}$, $g(\vec{e})\in L(\vec{e})$ is equivalent to finding an $f$-weighted $\beta^{\prime}$-orientation for some other $\mathbb{Z}_{2k+1}$-boundary $\beta^{\prime}$ of $G$, where the weight $f(e)$ of each edge $e$ is $2^{-1}$ times the difference between the two elements of $L(\vec{e})$. ∎

We now consider the case where $L(\vec{e})=\{0,1\}$ for every arc $\vec{e}\in\vec{E}$. Let $f_{2^{-1}}:\vec{E}\rightarrow\mathbb{Z}_{2k+1}$ denote the function that maps each arc $\vec{e}$ to $2^{-1}\pmod{2k+1}$. The same argument as in the proof of Lemma 9 implies that if ${G}$ has an $f_{2^{-1}}$-weighted $\beta$-orientation for every $\mathbb{Z}_{2k+1}$-boundary $\beta$, then for every $\mathbb{Z}_{2k+1}$-boundary $\beta$, the digraph $\vec{G}$ has a $\mathbb{Z}_{2k+1}$-flow $f$ with boundary $\beta$ such that $f(\vec{e})\in L(\vec{e})$ for every $\vec{e}$.

The following is a simple corollary of Theorem 6.

Corollary 10**.**

Let $\ell\geqslant 1$ be an odd integer and let $k\geqslant 1$ be relatively prime with $\ell$. Let $G=(V,E)$ be a $(3\ell-3)$-edge-connected graph, and let ${\mathbf{k}}:E\rightarrow\mathbb{Z}_{\ell}$ be the mapping that assigns $k\pmod{\ell}$ to each edge $e\in E$. Then for any $\mathbb{Z}_{\ell}$-boundary $\beta$, $G$ has a $\mathbf{k}$-weighted $\beta$-orientation.

Proof.

Observe that $\beta^{\prime}=k^{-1}\cdot\beta$ is a $\mathbb{Z}_{\ell}$-boundary ($k^{-1}$ is well defined in $\mathbb{Z}_{\ell}$). It follows from Theorem 6 that $G$ has a $\beta^{\prime}$-orientation. Note that this corresponds to a $\mathbf{k}$-weighted $\beta$-orientation of $G$, as desired. ∎

As a consequence, the following is an equivalent version of Theorem 6 (see also [12, 14]).

Theorem 11**.**

Let $k\geqslant 1$ be an integer and let $\vec{G}=(V,\vec{E})$ be a directed $6k$-edge-connected graph. Then for every $\mathbb{Z}_{2k+1}$-boundary $\beta$, $\vec{G}$ has a $\mathbb{Z}_{2k+1}$-flow $f$ with boundary $\beta$ such that $f(\vec{E})\in\{0,1\}\pmod{2k+1}$.

This version of Theorem 6 will allow us to derive interesting results on antisymmetric flows in directed highly edge-connected graphs. Given an abelian group $(B,+)$, a $B$-flow in $\vec{G}$ is a mapping $f:\vec{E}\rightarrow B$ such that $\partial f(v)=0$ for every vertex $v$, where all operations are performed in $B$. A $B$-flow $f$ in $\vec{G}=(V,\vec{E})$ is a nowhere-zero $B$-flow (or a $B$-nzf) if $0\not\in f(\vec{E})$, i.e. each arc of $\vec{G}$ is assigned a non-zero element of $B$. If no two arcs receive inverse elements of $B$, then $f$ is an antisymmetric $B$-flow (or a $B$-asf).

Since $0=-0$, a $B$-asf is also a $B$-nzf. It was conjectured by Tutte that every directed 2-edge-connected graph has a $\mathbb{Z}_{5}$-nzf [21], and that every directed 4-edge-connected graph has a $\mathbb{Z}_{3}$-nzf (see [18] and [3]). Antisymmetric flows were introduced by Nešetřil and Raspaud in [16]. A natural obstruction for the existence of an antisymmetric flow in a directed graph $\vec{G}$ is the presence of directed 2-edge-cut in $\vec{G}$. Nešetřil and Raspaud asked whether any directed graph without directed 2-edge-cut has a $B$-asf, for some $B$. This was proved by DeVos, Johnson, and Seymour in [7], who showed that any directed graph without directed 2-edge-cut has a $\mathbb{Z}_{2}^{8}\times\mathbb{Z}_{3}^{17}$-asf. It was later proved by DeVos, Nešetřil, and Raspaud [8], that the group could be replaced by $\mathbb{Z}_{2}^{6}\times\mathbb{Z}_{3}^{9}$. The best known result is due to Dvořák, Kaiser, Král’, and Sereni [10], who showed that any directed graph without directed 2-edge-cut has a $\mathbb{Z}_{2}^{3}\times\mathbb{Z}_{3}^{9}$-asf (this group has $157464$ elements).

Adding a stronger condition on the edge-connectivity allows to prove stronger results on the size of the group $B$. It was proved by DeVos, Nešetřil, and Raspaud [8], that every directed 4-edge-connected graph has a $\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{3}^{4}$-asf, that every directed 5-edge-connected graph has a $\mathbb{Z}_{3}^{5}$-asf, and that every directed 6-edge-connected graph has a $\mathbb{Z}_{2}\times\mathbb{Z}_{3}^{2}$-asf.

In [11], Jaeger conjectured the following weaker version of Tutte’s 3-flow conjecture: there is a constant $k$ such that every $k$-edge-connected graph has a $\mathbb{Z}_{3}$-nzf. This conjecture was recently solved by Thomassen [19], who proved that every 8-edge-connected graph has a $\mathbb{Z}_{3}$-nzf, and was improved by Lovász, Thomassen, Wu and Zhang [15], that every 6-edge-connected graph has a $\mathbb{Z}_{3}$-nzf (this is a simple consequence of Theorem 6).

The natural antisymmetric variant of Jaeger’s weak 3-flow conjecture would be the following: there is a constant $k$ such that every directed $k$-edge-connected graph has a $\mathbb{Z}_{5}$-asf.

Note that the size of the group would be best possible, since in $\mathbb{Z}_{2}$ and $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ every element is its own inverse, while a $\mathbb{Z}_{3}$-asf or a $\mathbb{Z}_{4}$-asf has to assign the same value to all the arcs (and this is impossible in the digraph on two vertices $u,v$ with exactly $k$ arcs directed from $u$ to $v$, for any integer $k\equiv 1\pmod{12}$).

Our final result is the following.

Theorem 12**.**

For any $k\geqslant 2$, every directed $\lceil\tfrac{6k}{k-1}\rceil$-edge-connected graph has a $\mathbb{Z}_{2k+1}$-asf.

Proof.

Let $k\geqslant 2$, and let $\vec{G}$ be a directed $\lceil\tfrac{6k}{k-1}\rceil$-edge-connected graph. Let $\vec{H}$ be the directed graph obtained from $\vec{G}$ by replacing every arc $\vec{e}$ by $k-1$ arcs with the same tail and head as $\vec{e}$, and let $H$ be the non-oriented graph underlying $\vec{H}$. Let $\beta(v)=d_{\vec{G}}^{-}(v)-d_{\vec{G}}^{+}(v)$ for every $v$. Since $\vec{G}$ is $\lceil\tfrac{6k}{k-1}\rceil$-edge-connected, $H$ is $6k$-edge-connected and by Theorem 11, $\vec{H}$ has a $\mathbb{Z}_{2k+1}$-flow $f$ with boundary $\beta$ with flow values in the set $\{0,1\}\pmod{2k+1}$. For any arc $\vec{e}$ of $\vec{G}$, let $g(\vec{e})$ be the sum of the values of the flow $f$ on the $t$ arcs corresponding to $\vec{e}$ in $\vec{H}$. Then $g$ is a $\mathbb{Z}_{2k+1}$-flow with boundary $\beta$ in $\vec{G}$, with flow values in the set $\{0,1,\ldots,k-1\}\pmod{2k+1}$. Now, set $g^{\prime}(\vec{e})=g(\vec{e})+1$ for every arc $\vec{e}$. Hence every $\vec{e}$ is assigned a value in $\{1,\ldots,k\}\pmod{2k+1}$, and $\partial g^{\prime}(v)\equiv\partial g(v)+d_{\vec{G}}^{+}(v)-d_{\vec{G}}^{-}(v)\equiv\beta^{\prime}(v)+d_{\vec{G}}^{+}(v)-d_{\vec{G}}^{-}(v)\equiv 0\pmod{2k+1}$ for every $v$. Thus $g^{\prime}$ is a $\mathbb{Z}_{2k+1}$-flow of $\vec{G}$ with flow values in the set $\{1,\ldots,k\}\pmod{2k+1}$, and thus a $\mathbb{Z}_{2k+1}$-asf in $\vec{G}$, as desired. This concludes the proof of Theorem 12. ∎

As a corollary, we directly obtain:

Corollary 13**.**

**

1. (i)

Every directed 7-edge-connected graph has a $\mathbb{Z}_{15}$-asf. 2. (ii)

Every directed 8-edge-connected graph has a $\mathbb{Z}_{9}$-asf. 3. (iii)

Every directed 9-edge-connected graph has a $\mathbb{Z}_{7}$-asf. 4. (iv)

Every directed 12-edge-connected graph has a $\mathbb{Z}_{5}$-asf.

By duality, using the results of Nešetřil and Raspaud [16], Corollary 13 (which, again, can be seen as an antisymmetric analogue of the statement of Jaeger’s conjecture) directly implies that every orientation of a planar graph of girth (length of a shortest cycle) at least 12 has a homomorphism to an oriented graph on at most 5 vertices. This was proved by Borodin, Ivanova and Kostochka in 2007 [4], and it is not known whether the same holds for planar graphs of girth at least 11. On the other hand, it was proved by Nešetřil, Raspaud and Sopena [17] that there are orientations of some planar graphs of girth at least 7 that have no homomorphism to an oriented graph of at most 5 vertices. By duality again, this implies that there are directed 7-edge-connected graphs with no $\mathbb{Z}_{5}$-asf. We conjecture the following:

Conjecture 14**.**

Every directed 8-edge-connected graph has a $\mathbb{Z}_{5}$-asf.

It was conjectured by Lai [13] that for every $k\geqslant 1$, every $(4k+1)$-edge-connected graph $G$ has a $\beta$-orientation for every $\mathbb{Z}_{2k+1}$-boundary $\beta$ of $G$. If true, this conjecture would directly imply (using the same proof as that of Theorem 12) that for any $k\geqslant 2$, every directed $\lceil\tfrac{4k+1}{k-1}\rceil$-edge-connected graph has a $\mathbb{Z}_{2k+1}$-asf. In particular, this would show that directed 5-edge-connected graph have a $\mathbb{Z}_{13}$-asf, directed 6-edge-connected graph have a $\mathbb{Z}_{9}$-asf, directed 7-edge-connected graph have a $\mathbb{Z}_{7}$-asf, and directed 9-edge-connected graph have a $\mathbb{Z}_{5}$-asf. The bound on directed 5-edge-connected graph would also directly imply, using the proof of the main result of [10], that directed graphs with no directed 2-edge-cut have a $\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{3}^{4}\times\mathbb{Z}_{13}$-asf.

3. Proof of Theorem 3

We first recall the following (weak form of a) classical result by Mader (see [9], Theorem 1.4.3):

Lemma 15**.**

Given an integer $k\geqslant 1$, if $G=(V,E)$ is a graph with average degree at least $4k$, then there is a subset $X$ of $V$ such that $|X|>1$ and $G[X]$ is $(k+1)$-edge-connected.

We will also need the following result of Thomassen [20], which is a simple consequence of Theorem 6.

Theorem 16** ([20]).**

Let $k\geqslant 3$ be an odd integer, $G=(V_{1},V_{2},E)$ be a bipartite graph, and $f:V_{1}\cup V_{2}\rightarrow\mathbb{Z}_{k}$ be a mapping satisfying $\sum_{v\in V_{1}}f(v)\equiv\sum_{v\in V_{2}}f(v)\pmod{k}$. If $G$ is $(3k-3)$-edge-connected, then $G$ has a spanning subgraph $H$ such that for any $v\in V$, $d_{H}(v)\equiv f(v)\pmod{k}$.

Let $G$ be a graph, and let $X$ and $Y$ be two disjoint subsets of vertices of $G$. The set of edges of $G$ with one endpoint in $X$ and the other in $Y$ is denoted by $E(X,Y)$.

We are now ready to prove Theorem 3.

Proof of Theorem 3. We proceed by induction on $n$. For $n=1$, this is a direct consequence of Theorem 1, so suppose that $n\geqslant 2$. Each basis $B_{s}$ can be considered as an $n\times n$ matrix where each column is a vector with support of size at most 2. Let $\mathcal{B}=\biguplus_{i=1}^{t}B_{i}$.

For $1\leqslant i\leqslant n$, a vector is called an $i$-vector if its support is the singleton $\{i\}$ (in other words, the $i$-th entry is non-zero and all the other entries are zero). Suppose that for some $1\leqslant i\leqslant n$, $\mathcal{B}$ contains at least $p-1$ $i$-vectors. Let $\mathcal{C}$ be the set of $i$-vectors of $\mathcal{B}$. Clearly, each basis contains at most one $i$-vector. For every $B_{s}$, let $B_{s}^{\prime}$ be the matrix obtained from $B_{s}$ by removing its $i$-vector (if any) and the $i^{th}$ row. Clearly $B_{s}^{\prime}$ is or contains a basis of $\mathbb{Z}_{p}^{n-1}$. By induction hypothesis, $\biguplus_{s=1}^{t}B_{s}^{\prime}$ forms an additive basis of $\mathbb{Z}_{p}^{n-1}$. In other words, for any vector $\beta=(\beta_{1},...,\beta_{i},...,\beta_{n})\in\mathbb{Z}_{p}^{n}$, there is a subset $Y_{1}$ of $\mathcal{B}\setminus\mathcal{C}$ which sums to $(\beta_{1},...,\hat{\beta}_{i},..,\beta_{n})$ for some $\hat{\beta}_{i}$. Since $|\mathcal{C}|\geqslant p-1$, it follows from Theorem 1 that there is a subset $Y_{2}$ of $\mathcal{C}$ which sums to $(0,...,\beta_{i}-\hat{\beta}_{i},..,0)$. Hence $Y_{1}\cup Y_{2}$ sums to $\beta$.

Thus we can suppose that there are at most $p-2$ $i$-vectors for every $i$. Then there are at least $8\ell(3p-4)n$ vectors with a support of size 2 in $\mathcal{B}$. Since there are at most $\ell$ distinct shadows of size 2 in $\mathcal{B}$, there are at least $8(3p-4)n$ vectors with the same (unordered) shadow of size 2, say $\{a_{1},a_{2}\}$ (recall that shadows are multisets, so $a_{1}$ and $a_{2}$ might coincide).

Let $G$ be the graph (recall that graphs in this paper are allowed to have multiple edges) with vertex set $V=\{v_{1},...,v_{n}\}$ and edge set $E$, where edges $v_{i}v_{j}$ are in one-to-one correspondence with vectors of $\mathcal{B}$ with support $\{i,j\}$ and shadow $\{a_{1},a_{2}\}$. Then $G$ contains at least $8(3p-4)n$ edges.

We now consider a random partition of $V$ into 2 sets $V_{1},V_{2}$ (by assigning each vertex of $V$ uniformly at random to one of the sets $V_{k}$, $k=1,2$). Let $e=v_{i}v_{j}$ be some edge of $G$. Recall that $e$ corresponds to some vector with only two non-zero entries, say without loss of generality $a_{1}$ at $i^{th}$ index and $a_{2}$ at $j^{th}$ index. The probability that $v_{i}$ is assigned to $V_{1}$ and $v_{j}$ is assigned to $V_{2}$ is at least $\tfrac{1}{4}$. As a consequence, there is a partition of $V$ into 2 sets $V_{1},V_{2}$ and a subset $E^{\prime}\subseteq E(V_{1},V_{2})$ of at least $8(3p-4)n/4=2(3p-4)n$ edges such that for every $e\in E^{\prime}$, the vector of $\mathcal{B}$ corresponding with $e$ has entry $a_{1}$ (resp. $a_{2}$) at the index associated to the endpoint of $e$ in $V_{1}$ (resp. $V_{2}$).

Since the graph $G^{\prime}=(V,E^{\prime})$ has average degree at least $4(3p-4)$, it follows from Lemma 15 that there is a set $X\subseteq V$ of at least 2 vertices, such that $G^{\prime}[X]$ is $(3p-3)$-edge-connected. Set $H=G^{\prime}[X]$ and $F$ the edge set of $H$. Note that $H$ is bipartite with bipartition $X_{1}=X\cap V_{1}$ and $X_{2}=X\cap V_{2}$.

For each integer $1\leqslant s\leqslant t$, let $B_{s}^{*}$ be the matrix obtained from $B_{s}$ by doing the following: for each vertex $v_{i}$ in $X_{1}$ (resp. $X_{2}$), we multiply all the elements of the $i^{th}$ row of $B_{s}$ by $a_{1}^{-1}$ (resp. $-a_{2}^{-1}$), noting that all the operations are performed in $\mathbb{Z}_{p}$. Let $\mathcal{B}^{*}=\biguplus_{s=1}^{t}B_{s}^{*}$. Note that each vector of $\mathcal{B}^{*}$ corresponding to some edge $e\in F$ has shadow $\{1,-1\}$ ($1$ is the entry indexed by the endpoint of $e$ in $X_{1}$ and $-1$ is the entry indexed by the endpoint of $e$ in $X_{2}$). It is easy to verify the following.

• •

Each $B_{s}^{*}$ is a linear basis of $\mathbb{Z}_{p}^{n}$.

• •

$\mathcal{B}$ is an additive basis if and only if $\mathcal{B}^{*}$ is an additive basis.

Hence it suffices to prove that $\mathcal{B}^{*}$ is an additive basis.

Without loss of generality, suppose that $X=\{v_{m},...,v_{n}\}$ for some $m\leqslant n-1$. By contracting $k$ rows of a matrix, we mean deleting these $k$ rows and adding a new row consisting of the sum of the $k$ rows. For each $1\leqslant s\leqslant t$, let $B_{s}^{\prime}$ be the matrix of $m$ rows obtained from $B_{s}^{*}$ by contracting all $m^{th},(m+1)^{th},...,n^{th}$ rows. Note that the operation of contracting $k$ rows decreases the rank of the matrix by at most $k-1$ (since it is the same as replacing one of the rows by the sum of the $k$ rows, which preserves the rank, and then deleting the $k-1$ other rows). Let $\mathcal{B}^{\prime}=\biguplus_{s=1}^{t}B_{s}^{\prime}$ . Since each $B_{s}^{*}$ is a linear basis of $\mathbb{Z}_{p}^{n}$, each $B_{s}^{\prime}$ has rank at least $m$ and therefore contains a basis of $\mathbb{Z}_{p}^{m}$. Hence, by induction hypothesis, $\mathcal{B}^{\prime}\setminus\mathcal{B}_{0}^{\prime}$ is an additive basis of $\mathbb{Z}_{p}^{m}$, where $\mathcal{B}_{0}^{\prime}$ is the set of all columns with empty support in $\mathcal{B}^{\prime}$. For every $\beta=(\beta_{1},...,\beta_{n})\in\mathbb{Z}_{p}^{n}$, let $\beta^{\prime}=(\beta_{1},...,\beta_{m-1},\sum_{i=m}^{n}\beta_{i})\in\mathbb{Z}_{p}^{m}$. Then there is a subset $Y^{\prime}$ of $\mathcal{B}^{\prime}\setminus\mathcal{B}_{0}^{\prime}$ which sums to $\beta^{\prime}$. Let $Y^{*}$ and $\mathcal{B}_{0}^{*}$ be the subsets of $\mathcal{B}^{*}$ corresponding to $Y^{\prime}$ and $\mathcal{B}_{0}^{\prime}$, respectively. Then $Y^{*}$ sums to some $\hat{\beta}=(\beta_{1},...,\beta_{m-1},\hat{\beta}_{m},...,\hat{\beta}_{n})$, where $\sum_{i=m}^{n}\hat{\beta}_{i}\equiv\sum_{i=m}^{n}\beta_{i}\pmod{p}$.

Recall that for each edge $e\in F$, the corresponding vector in $\mathcal{B}^{*}$ has precisely two non-zero entries, $(1,-1)$, each with index in $X$. Hence the vector corresponding to each $e\in F$ in $\mathcal{B}^{\prime}$ has empty support. Thus the set of vectors in $\mathcal{B}^{*}$ corresponding to the edge set $F$ is a subset of $\mathcal{B}_{0}^{*}$, which is disjoint from $Y$.

For each $v_{i}\in X_{1}$, let $\beta_{X}(v_{i})=\beta_{i}-\hat{\beta}_{i}$, and for each $v_{i}\in X_{2}$, let $\beta_{X}(v_{i})=\hat{\beta}_{i}-\beta_{i}$. Since $\sum_{i=m}^{n}\hat{\beta}_{i}\equiv\sum_{i=m}^{n}\beta_{i}\pmod{p}$, we have $\sum_{v_{i}\in X\cap V_{1}}\beta_{X}(v_{i})=\sum_{v_{i}\in X\cap V_{2}}\beta_{X}(v_{i})$. Since $H$ is $(3p-3)$-edge-connected, it follows from Theorem 16 that there is a subset $F^{\prime}\subseteq F$ such that, in the graph $(X,F^{\prime})$, each vertex $v_{i}\in X_{1}$ has degree $\beta_{i}-\hat{\beta}_{i}\pmod{p}$ and each vertex $v_{i}\in X_{2}$ has degree $\hat{\beta}_{i}-\beta_{i}\pmod{p}$. Therefore, $F^{\prime}$ corresponds to a subset $Z^{*}$ of vectors of $\mathcal{B}_{0}^{*}$, summing to $(0,\ldots,0,\beta_{m}-\hat{\beta}_{m},\ldots,\beta_{n}-\hat{\beta}_{n})$. Then $Y^{*}\cup Z^{*}$ sums to $\beta$. It follows that $\mathcal{B}^{*}$ is an additive basis of $\mathbb{Z}_{p}^{n}$, and so is $\mathcal{B}$. This completes the proof. $\Box$

4. Two proofs of (versions of) Theorem 7

We now give two proofs of (versions of) Theorem 7. The first one is a direct application of Corollary 5, but requires a stronger assumption on the edge-connectivity of $G$ ($24p^{2}-54p+28$ instead of $6p^{2}-14p+8$ for the second proof).

First proof of Theorem 7. We fix some arbitrary orientation $\vec{G}=(V,\vec{E})$ of $G$ and denote the vertices of $G$ by $v_{1},\ldots,v_{n}$. The number of edges of $G$ is denoted by $m$. For each arc $\vec{e}=(v_{i},v_{j})$ of $\vec{G}$, we associate $\vec{e}$ to a vector $x_{e}\in(\mathbb{Z}_{p}^{n})_{0}$ in which the $i^{th}$-entry is equal to $f(e)\pmod{p}$, the $j^{th}$-entry is equal to $-f(e)\pmod{p}$ and all the remaining entries are equal to $0\pmod{p}$.

Let us consider the following statements.

1. (a)

For each $\mathbb{Z}_{p}$-boundary $\beta$, there is an $f$-weighted $\beta$-orientation of $G$. 2. (b)

For each $\mathbb{Z}_{p}$-boundary $\beta$ there is a vector $(a_{e})_{e\in E}\in\{-1,1\}^{m}$, such that $\sum_{e\in E}a_{e}x_{e}\equiv\beta\pmod{p}$. 3. (c)

For each $\mathbb{Z}_{p}$-boundary $\beta$ there is a vector $(a_{e})_{e\in E}\in\{0,1\}^{m}$ such that $\sum_{e\in E}2a_{e}x_{e}\equiv\beta\pmod{p}$.

Clearly, a is equivalent to b. We now claim that b is equivalent to c. To see this, simply do the following for each arc $\vec{e}=(v_{i},v_{j})$ of $\vec{G}$: add $f(e)$ to the $j^{th}$-entry of $x_{e}$ and to $\beta(v_{j})$, and subtract $f(e)$ from the $i^{th}$-entry of $x_{e}$ and from $\beta(v_{i})$. To deduce c from Corollary 5, what is left is to show that $\{a_{e}:e\in E\}$ can be decomposed into sufficiently many linear bases of $(\mathbb{Z}_{p}^{n})_{0}$. This follows from the fact that $G$ is $(8(p-1)(3p-4)+2p-4)$-edge-connected (and therefore contains $4(p-1)(3p-4)+p-2$ edge-disjoint spanning trees) and that the set of vectors $a_{e}$ corresponding to the edges of a spanning tree of $G$ forms a linear basis of $(\mathbb{Z}_{p}^{n})_{0}$ (see [12]). $\Box$

A second proof consists in mimicking the proof of Theorem 3 (it turns out to give a better bound for the edge-connectivity of $G$).

Second proof of Theorem 7. As before, all values and operations are considered modulo $p$. We can assume without loss of generality that $f(E)\in\{1,2,\ldots,\tfrac{p-1}{2}\}$, since otherwise we can replace the value $f(e)$ of an edge $e$ by $-f(e)$, without changing the problem.

We prove the result by induction on the number of vertices of $G$. The result is trivial if $G$ contains only one vertex, so assume that $G$ has at least two vertices.

For any $1\leqslant i\leqslant k$, let $E_{i}$ be the set of edges $e\in E$ with $f(e)=i$, and let $G_{i}=(V,E_{i})$. Since $G$ is $(6p-8)(p-1)$-edge-connected, $G$ has minimum degree at least $(6p-8)(p-1)$ and then average degree at least $(6p-8)(p-1)$. As a consequence, there exists $i$ such that $G_{i}$ has average degree at least $12p-16$. By Lemma 15, since $\tfrac{12p-16}{4}+1=3p-3$, $G_{i}$ has an induced subgraph $H=(X,F)$ with at least two vertices such that $H$ is $(3p-3)$-edge-connected. Let $G/X$ be the graph obtained from $G$ by contracting $X$ into a single vertex $x$ (and removing possible loops). Since $H$ contains more than one vertex, $G/X$ has less vertices than $G$ (note that possibly, $X=V$ and in this case $G/X$ consists of the single vertex $x$). Since $G$ is $(6p-8)(p-1)$-edge-connected, $G/X$ is also $(6p-8)(p-1)$-edge-connected. Hence by the induction hypothesis it has an $f$-weighted $\beta$-orientation, where we consider the restriction of $f$ to the edge-set of $G/X$, and we define $\beta(x)=\beta(X)$. Note that this orientation corresponds to an orientation of all the edges of $G$ with at most one endpoint in $X$.

We now orient arbitrarily the edges of $G[X]$ not in $F$ (the edge-set of $H$), and update the values of the $\mathbb{Z}_{p}$-boundary $\beta$ accordingly (i.e. for each $v\in X$, we subtract from $\beta(v)$ the contribution of the arcs that were already oriented). It is easy to see that as the original $\beta$ was a boundary, the new $\beta$ is indeed a boundary. Finally, since all the edges of $H$ have the same weight, and since $H$ is $(3p-3)$-edge-connected, it follows from Corollary 10 that $H$ has an $f$-weighted $\beta$-orientation (with respect to the updated boundary $\beta$). The orientations combine into an $f$-weighted $\beta$-orientation of $G$, as desired. $\Box$

Acknowledgments

We would like to thank the referees for their suggestions and for mentioning the existence of the unpublished manuscript [14].