On Alon-Tarsi orientations of sparse graphs

TL;DR

This paper investigates Alon-Tarsi orientations in sparse graphs, extending known results from outerplanar to $K_4$-minor-free graphs and characterizing extendability for specific vertex triples and pairs.

Contribution

It strengthens existing results by proving $(2,2)$-AT extendability for $K_4$-minor-free graphs and characterizes all triples and pairs for which certain extendability properties hold.

Findings

$K_4$-minor-free graphs are $(2,2)$-AT extendable with respect to any two vertices.

Complete characterization of triples $(x,y,z)$ for $(2,2,2)$-AT extendability.

Characterization of pairs $(x,y)$ for $(2,1)$-AT extendability in $K_4$-minor-free graphs.

Abstract

Assume $G$ is a graph, $(v_1,\ldots,v_k)$ is a sequence of distinct vertices of $G$, and $(a_1,\ldots,a_k)$ is an integer sequence with $a_i \in \{1,2\}$. We say $G$ is \emph{$(a_1,\ldots,a_k)$-list extendable} (respectively, \emph{$(a_1,\ldots,a_k)$-AT extendable}) with respect to $(v_1,\ldots,v_k)$ if $G$ is $f$-choosable (respectively, $f$-AT), where $f(v_i)=a_i $ for $i \in \{1,\ldots, k\}$, and $f(v)=3$ for $v \in V(G) \setminus \{v_1,\ldots, v_k\}$. Hutchinson proved that if $G$ is an outerplanar graph, then $G$ is $(2,2)$-list extendable with respect to $(x,y)$ for any vertices $x,y$. We strengthen this result and prove that if $G$ is a $K_4$-minor-free graph, then $G$ is $(2,2)$-AT extendable with respect to $(x,y)$ for any vertices $x,y$. Then we characterize all triples $(x,y,z)$ of a $K_4$-minor-free graph $G$ for which $G$ is $(2,2,2)$-AT extendable (as well as $(2,2,2)$-list extendable) with respect to $(x,y,z)$. We also characterize the pairs $(x,y)$ of a $K_4$-minor-free graph $G$ for which $G$ is $(2,1)$-AT extendable (as well as $(2,1)$-list extendable) with respect to $(x,y)$. Moreover, we characterize all triples $(x,y,z)$ of a 3-colorable graph $G$ with its maximum average degree less than $\frac{14}{5}$ for which $G$ is $(2,2,2)$-AT extendable with respect to $(x,y,z)$.

Equations

$$\mathrm{diff}(D)=|\mathrm{EE}(D)|-|\mathrm{OE}(D)|.$$

$$V(D_n)=\{u_i,v_i,w_i:i\in [n]\}\cup \{u_0\},\text{hfill}E(D_n)=\{u_{i-1}{v}_i,u_{i-1}{w}_i,v_i{w}_i,v_i{u}_i,w_i{u}_i:i\in [n]\}.$$

$$\mathrm{diff}(D)=\mathrm{diff}(D^{\prime }).$$

$$0>5\mathrm{mad}(G)|V(G)|-14|V(G)|\ge \sum _{v\in V(G)}(5d_G(v)-14)\ge -4n_2+n_3+6n_4+11n_5^{+}.$$

$$12\ge 4|B|=4n_2>n_3+6n_4+11n_5^{+}\ge n_3^{+}=|V(G)\setminus B|\ge 5-|B|.$$

$$12\ge 4|B|\ge n_3+6n_4+2.$$

Full text

1 Introduction

Let $\mathbb{N}$ denote the set of positive integers, and for $k\in\mathbb{N}$, define $[k]$ to be the set $\{1,\ldots,k\}$. Throughout the paper, we assume that $G$ is a simple graph unless stated otherwise. Given $G$, let $V(G)$ and $E(G)$ denote its vertex set and edge set, respectively. For an integer $k$, a vertex $v$ of $G$ is a $k$-vertex (respectively, a $k^{+}$-vertex or a $k^{-}$-vertex) if $d_{G}(v)=k$ (respectively, $d_{G}(v)\geq k$ or $d_{G}(v)\leq k$). A proper coloring of $G$ is a function $\phi:V(G)\to\mathbb{N}$ such that $\phi(u)\neq\phi(v)$ for each edge $uv$ of $G$. Given $k\in\mathbb{N}$, we say $G$ is $k$-colorable if $G$ has a proper coloring $\phi$ such that $\phi(V(G))\subseteq[k]$. A list assignment $L$ of $G$ is a function on $V(G)$ that assigns a list $L(v)\subseteq\mathbb{N}$ of available colors to each vertex $v\in V(G)$. Given a list assignment $L$ of $G$, an $L$-coloring $\varphi$ of $G$ is a proper coloring of $G$ such that $\varphi(v)\in L(v)$ for each vertex $v\in V(G)$. Let $\mathbb{N}^{G}$ be the set of all mappings $f:V(G)\to\mathbb{N}$. For a mapping $f\in\mathbb{N}^{G}$, we say $G$ is $f$-choosable if $G$ has an $L$-coloring for every list assignment $L$ of $G$ for which $|L(v)|\geq f(v)$. If $f$ is a constant function with value $k\in\mathbb{N}$, then we say an $f$-choosable graph $G$ is $k$-choosable. The list chromatic number of $G$, denoted $\chi_{l}(G)$, is the minimum $k$ such that $G$ is $k$-choosable. List coloring of graphs has been studied extensively in the literature [6]. A useful tool in the study of list coloring is the Combinatorial Nullstellensatz, and its associated Alon-Tarsi orientations of graphs.

Definition 1.1**.**

Let $D$ be an orientation (of edges) of $G$. An Eulerian sub-digraph of $D$ is a spanning sub-digraph $F$ of $D$ with $d_{F}^{+}(v)=d_{F}^{-}(v)$ for every vertex $v\in V(G)$. Let $\operatorname{EE}(D)$ (respectively, $\operatorname{OE}(D)$) denote the set of Eulerian sub-digraphs with an even (respectively, odd) number of arcs. Let

[TABLE]

We say $D$ is an Alon-Tarsi orientation (shortened as AT-orientation) if $\operatorname{diff}(D)\neq 0$. For a mapping $f\in\mathbb{N}^{G}$, we say $G$ is $f$-Alon-Tarsi (shortened as $f$-AT) if $G$ has an AT-orientation $D$ with $d_{D}^{+}(v)\leq f(v)-1$ for each vertex $v\in V(G)$. If $f$ is a constant function with value $k\in\mathbb{N}$, then we say an $f$-AT graph $G$ is $k$-AT. The Alon-Tarsi number of $G$, denoted $AT(G)$, is the minimum integer $k$ such that $G$ is $k$-AT.

Alon-Tarsi Theorem** ([1]).**

If $G$ is $f$-AT, then $G$ is $f$-choosable. In particular, $\chi_{l}(G)\leq AT(G)$.

The Alon-Tarsi number of a graph $G$ is not only an upper bound for both the list chromatic number of $G$ and the online list chromatic number of $G$ [9], but also a graph invariant of independent interest. A natural question is whether some upper bounds for the list chromatic number of graphs are also upper bounds for the Alon-Tarsi number. A classical result of Thomassen [5] says that every planar graph is $5$-choosable. This result was strengthened in [8], where it was proved that every planar graph has Alon-Tarsi number at most $5$. Indeed, Thomassen’s classical result is stronger than the statement that every planar graph is $5$-choosable: if $G$ is a plane graph with boundary cycle $(v_{1}v_{2}\ldots v_{n})$, then $G-v_{1}v_{2}$ is $f$-choosable, where $f(v_{1})=f(v_{2})=1$, $f(v_{i})=3$ for $i\in\{3,\ldots,n\}$, and $f(v)=5$ for each interior vertex $v$. This stronger and more technical result is useful in many cases, say for example in the study of choosability of locally planar graphs [2]. The result in [8] also says that $G-v_{1}v_{2}$ is $f$-AT for the same aforementioned function $f$.

In this paper, we are interested in list colorings and Alon-Tarsi orientations of $K_{4}$-minor-free graphs. It is well-known and easy to verify that $K_{4}$-minor-free graphs are $2$-degenerate, and hence has list chromatic number, as well as Alon-Tarsi number, at most $3$. We are interested in the problem whether a $K_{4}$-minor-free graph $G$ is $f$-choosable, or $f$-AT, for some $f\in\mathbb{N}^{G}$, with $f(v)\leq 3$ for every vertex $v$ of $G$, and $f(v)<3$ for some vertices $v$ of $G$.

Definition 1.2**.**

Assume $G$ is a graph, $(v_{1},\ldots,v_{k})$ is a $k$-tuple of distinct vertices of $G$, and $(a_{1},\ldots,a_{k})$ is a sequence of integers with $a_{i}\in[2]$ for $i\in[k]$. Define $f\in\mathbb{N}^{G}$ as $f(v_{i})=a_{i}$ for $i\in[k]$ and $f(v)=3$ for $v\in V(G)\setminus\{v_{1},\ldots,v_{k}\}$. If $G$ is $f$-choosable (respectively, $f$-AT), then we say $G$ is $(a_{1},\ldots,a_{k})$-list extendable (respectively, $(a_{1},\ldots,a_{k})$-AT extendable) with respect to $(v_{1},\ldots,v_{k})$. An $f$-AT orientation of $G$ is called an $(a_{1},\ldots,a_{k})$-AT orientation of $G$ with respect to $(v_{1},\ldots,v_{k})$.

Hutchinson [4] first studied $f$-choosability of outerplanar graphs. Hutchinson proved that all outerplanar graphs are $(2,2)$-list extendable with respect to any pair of vertices $(x,y)$, and presented necessary and sufficient conditions for an outerplanar graph $G$ to be $(2,1)$-list extendable or $(1,1)$-list extendable with respect to $(x,y)$.

The results in this paper extend Hutchinson’s results in two aspects: (1) we consider a more general class of graphs, from outerplanar graphs to $K_{4}$-minor-free graphs, or graphs with bounded maximum average degree and (2) prove stronger statements, from list extendability to AT extendability. More precisely, we first extend Hutchinson’s result to $K_{4}$-minor-free graphs, and strengthen the $(a,b)$-list extendable results to $(a,b)$-AT extendable results. Then for an arbitrary $K_{4}$-minor-free graph $G$, we characterize all triples $(x,y,z)$ for which $G$ is $(2,2,2)$-list extendable, as well as $(2,2,2)$-AT extendable. Lastly, we discuss a similar question in the context of graphs with bounded maximum average degree.

1.1 $(2,2)$-AT extendability of $K_{4}$-minor-free graphs

To state our result, we need more definitions. For $n\in\mathbb{N}$, let $D_{n}$ be the graph with

[TABLE]

For $n\in\mathbb{N}$, the graph $D_{n}$ is called a chain of diamonds (see Figure 1). Let $U(D_{n})$ denote the set $\{u_{0},u_{1},\ldots,u_{n}\}$ of vertices of $D_{n}$.

Definition 1.3**.**

Let $G$ be a graph. A set $X$ of distinct vertices of $G$ is said to be connected by a chain of diamonds if there is $n\in\mathbb{N}$ and a homomorphism $\varphi$ from a chain of diamonds $D_{n}$ to $G$ such that $X\subseteq\varphi(U(D_{n}))$.

Observation 1.4**.**

Observe that if $\phi$ is a proper $3$-coloring of a chain of diamonds $D_{n}$, then $\phi(u)=\phi(u^{\prime})$ for all $u,u^{\prime}\in U(D_{n})$. Hence if a subset $X$ of vertices of a $3$-colorable graph $G$ is connected by a chain of diamonds, then all vertices in $X$ are colored by the same color in every proper 3-coloring of $G$.

Our first result extends the work of [4] to AT-extendability of $K_{4}$-minor-free graphs.

Theorem 1.5**.**

Assume $G$ is a $K_{4}$-minor-free graph and $x,y$ are distinct vertices of $G$. Then $G$ is $(2,2)$-AT extendable with respect to $(x,y)$. Moreover, if $\{x,y\}$ is not connected by a chain of diamonds, then $G$ is $(2,1)$-AT extendable with respect to $(x,y)$.

The following corollary holds from Theorem 1.5.

Corollary 1.6**.**

Assume $G$ is a $K_{4}$-minor-free graph and $x,y$ are distinct vertices of $G$. Then the following are equivalent:

1. (1)

$G$* is $(2,1)$-AT extendable with respect to $(x,y)$.* 2. (2)

$G$* is $(2,1)$-list extendable with respect to $(x,y)$.* 3. (3)

$\{x,y\}$* is not connected by a chain of diamonds.*

Proof.

(3) $\Rightarrow$ (1) follows from Theorem 1.5, and (1) $\Rightarrow$ (2) follows from the Alon-Tarsi Theorem. To show (2) $\Rightarrow$ (3), assume $G$ is $(2,1)$-list extendable with respect to $(x,y)$. Let $L(x)=\{1,2\}$, $L(y)=\{3\}$, and $L(v)=\{1,2,3\}$ for all $v\in V(G)\setminus\{x,y\}$. Then $G$ has an $L$-coloring $f$, which is a proper 3-coloring of $G$ with $f(x)\neq f(y)$. By 1.4, $\{x,y\}$ is not connected by a chain of diamonds. ∎

1.2 $(2,2,2)$-AT extendability of $K_{4}$-minor-free graphs

Our second result considers $(2,2,2)$-AT extendability of $K_{4}$-minor-free graphs and prove the following result.

Definition 1.7**.**

A set $X$ of three distinct vertices of $G$ is feasible if $X$ is not connected by a chain of diamonds and there is a proper $3$-coloring $\phi$ of $G$ for which $|\phi(X)|\leq 2$.

Theorem 1.8**.**

Assume $G$ is a $K_{4}$-minor-free graph and $x,y,z$ are distinct vertices of $G$. If $\{x,y,z\}$ is feasible, then $G$ is $(2,2,2)$-AT extendable with respect to $(x,y,z)$.

As a corollary of Theorem 1.8, the following holds.

Corollary 1.9**.**

Assume $G$ is a $K_{4}$-minor-free graph and $x,y,z$ are distinct vertices of $G$. Then the following are equivalent:

1. (1)

$G$* is $(2,2,2)$-AT extendable with respect to $(x,y,z)$.* 2. (2)

$G$* is $(2,2,2)$-list extendable with respect to $(x,y,z)$.* 3. (3)

$\{x,y,z\}$* is feasible.*

Proof.

(3) $\Rightarrow$ (1) follows from Theorem 1.8, and (1) $\Rightarrow$ (2) follows from the Alon-Tarsi Theorem. To show (2) $\Rightarrow$ (3), assume $G$ is $(2,2,2)$-list extendable with respect to $(x,y,z)$. Let $L(x)=L(y)=L(z)=\{1,2\}$ and $L(v)=\{1,2,3\}$ for all $v\in V(G)\setminus\{x,y,z\}$. Then $G$ has an $L$-coloring $\varphi$, which is a proper $3$-coloring of $G$ such that $|\{\varphi(x),\varphi(y),\varphi(z)\}|\leq 2$. Let $L^{\prime}(x)=\{1,2\}$, $L^{\prime}(y)=\{1,3\}$, $L^{\prime}(z)=\{2,3\}$, and $L^{\prime}(v)=\{1,2,3\}$ for all $v\in V(G)\setminus\{x,y,z\}$. Again $G$ has an $L^{\prime}$-coloring $\phi$, which is a proper $3$-coloring of $G$ that uses at least two colors on $\{x,y,z\}$. By 1.4, $\{x,y,z\}$ is not connected by a chain of diamonds. Hence, the set $\{x,y,z\}$ is feasible. ∎

Theorem 1.8 is tight in the sense that there are $K_{4}$-minor-free graphs $G$ such that $G$ is not $(2,2,1)$-list extendable with respect to $(x,y,z)$ for any distinct $x,y,z\in V(G)$. Indeed, any $K_{4}$-minor-free graph with a unique proper $3$-coloring has this property.

Here is a sketch of proof. Let $G$ be a $K_{4}$-minor-free graph with a unique proper $3$-coloring $\phi$. If $\phi(x)=\phi(z)$, then let $L(x)=L(y)=\{1,2\}$, $L(z)=\{3\}$, and $L(v)=\{1,2,3\}$ for any other vertex $v$. It is easy to see that $G$ is not $L$-colorable, as any proper 3-coloring of $G$ colors $x,z$ with the same color. The case where $\phi(y)=\phi(z)$ is symmetric.

Assume $\phi(z)\not\in\{\phi(x),\phi(y)\}$. If $\phi(x)\neq\phi(y)$, then let $L(x)=L(y)=\{1,2\}$, $L(z)=\{1\}$, and $L(v)=\{1,2,3\}$ for any other vertex $v$. Then $G$ is not $L$-colorable. If $\phi(x)=\phi(y)$, then let $L(x)=\{1,2\},L(y)=\{1,3\},L(z)=\{1\}$, and $L(v)=\{1,2,3\}$ for any other vertex $v$. Again $G$ is not $L$-colorable.

In particular, $2$-trees are $K_{4}$-minor-free graphs that have a unique proper $3$-coloring. So they are not $(2,2,1)$-list extendable with respect to $(x,y,z)$ for any distinct vertices $x,y,z$.

Despite the above observation, if a $K_{4}$-minor-free graph is triangle-free, then we have the following result.

Theorem 1.10**.**

If $G$ is a triangle-free $K_{4}$-minor-free graph, then $G$ is $(2,2,1)$-AT extendable with respect to $(x,y,z)$ for every $x,y,z\in V(G)$.

Proof.

Let $G$ be a triangle-free $K_{4}$-minor-free graph. If $|V(G)|\leq 3$, then Theorem 1.10 holds. Thus, we may assume that $|V(G)|\geq 4$. Note that $G$ has at least four $2^{-}$-vertices since it is a triangle-free $K_{4}$-minor-free graph.

Suppose to the contrary that there is a graph $G$ and its vertices $x,y,z\in V(G)$ such that $G$ is not $(2,2,1)$-AT extendable with respect to $(x,y,z)$. Let $G$ be the minimal graph with this property with respect to $|V(G)|$. Let $w\in V(G)\setminus\{x,y,z\}$ such that $w$ is a $2^{-}$-vertex. By the minimality of $G$, $G-w$ has a $(2,2,1)$-AT orientation $D^{\prime}$ with respect to $(x,y,z)$. Then we obtain an orientation $D$ of $G$ by starting with $D^{\prime}$ and then orienting the edges incident with $w$ so that $d_{D}^{-}(w)=0$. Then $D$ is a $(2,2,1)$-AT orientation of $G$, which is a contradiction. ∎

1.3 $(2,2,2)$-AT extendability of graphs with bounded maximum average degree

Definition 1.11**.**

A set of three distinct vertices $\{x,y,z\}$ of $G$ is blocked if either for every proper 3-coloring $\phi$ of $G$, $|\phi(\{x,y,z\})|=3$ or for every proper 3-coloring $\phi$ of $G$, $|\phi(\{x,y,z\})|=1$.

Note that $G$ is not $(2,2,2)$-list extendable with respect to $(x,y,z)$ if $\{x,y,z\}$ is blocked. By using Corollary 1.9, it is easy to check that if $G$ is $K_{4}$-minor-free and $\{x,y,z\}$ is non-blocked, then $G$ is $(2,2,2)$-list extendable with respect to $(x,y,z)$. Figure 2 shows that the condition that $G$ be $K_{4}$-minor-free cannot be simply dropped. Nevertheless, we prove that if $G$ has ${\rm mad}(G)<\frac{14}{5}$, then $G$ is $(2,2,2)$-list extendable with respect to $(x,y,z)$, provided that $\{x,y,z\}$ is non-blocked; recall that the maximum average degree of $G$, denoted ${\rm mad}(G)$, is defined as $\max_{H\subseteq G}{2|E(H)|\over|V(H)|}$. Note that the graph in Figure 2 has ${\rm mad}(G)=\frac{14}{5}$.

Theorem 1.12**.**

If $G$ is a graph with $\operatorname{mad}(G)<{14\over 5}$, then $G$ is $(2,2,2)$-AT extendable with respect to $(x,y,z)$ for every $\{x,y,z\}$ that is non-blocked.

2 Preliminaries

If $u$ is a cut-vertex of a graph $G$, and $G_{1},G_{2}$ are two induced connected subgraphs of $G$ with $V(G_{1})\cap V(G_{2})=\{u\}$ and $V(G_{1})\cup V(G_{2})=V(G)$, then we say $u$ separates $G$ into $G_{1}$ and $G_{2}$. For an orientation $D$ of $G$, let $A(D)$ denote its arc set.

Lemma 2.1**.**

For a graph $G$, let $u$ be a cut-vertex (of $G$) that separates $G_{1}$ and $G_{2}$. For an orientation $D$ of $G$, let $D_{i}$ be the orientation of $G_{i}$ obtained by restricting $D$ onto $G_{i}$ for $i\in[2]$. Then $\operatorname{diff}(D)=\operatorname{diff}(D_{1})\times\operatorname{diff}(D_{2})$.

Proof.

For an orientation $D$ of $G$, let $D_{i}$ be the orientation of $G_{i}$ obtained by restricting $D$ onto $G_{i}$ for $i\in[2]$. Let $F$ be an Eulerian sub-digraph of $D$, and let $F_{i}$ be the sub-digraph of $D_{i}$ obtained by restricting $F$ onto $D_{i}$ for $i\in[2]$. Note that $\sum_{v\in V(F_{1})}d^{+}_{F_{1}}(v)=\sum_{v\in V(F_{1})}d^{-}_{F_{1}}(v)$, and $d^{+}_{F_{1}}(v)=d^{-}_{F_{1}}(v)$ for all $v\in V(F_{1})\setminus\{u\}$. Thus, $d^{+}_{F_{1}}(u)=d^{-}_{F_{1}}(u)$ and therefore $F_{1}$ is an Eulerian sub-digraph of $D_{1}$. Similarly, $F_{2}$ is an Eulerian sub-digraph of $D_{2}$. So $F$ is the disjoint (with respect to arcs) union of $F_{1}$ and $F_{2}$. Conversely, for any Eulerian sub-digraph $F_{1}$ of $D_{1}$, and any Eulerian sub-digraph $F_{2}$ of $D_{2}$, $F=F_{1}\cup F_{2}$ is an Eulerian sub-digraph of $D$. Note that $|A(F)|$ is even if $|A(F_{1})|$ and $|A(F_{2})|$ have the same parity, and $|A(F)|$ is odd if $|A(F_{1})|$ and $|A(F_{2})|$ have different parities. Hence $\operatorname{diff}(D)=\operatorname{diff}(D_{1})\times\operatorname{diff}(D_{2})$. ∎

Lemma 2.2**.**

Assume $G$ is a graph, $[u_{1}u_{2}u_{3}]$ is a triangle, $d_{G}(u_{1})=2$, and $d_{G}(u_{2})=3$ with $N_{G}(u_{2})=\{u_{1},u_{3},u_{4}\}$. Let $D$ be an orientation of $G$ in which the edges incident with $u_{1}$ or $u_{2}$ are oriented as $(u_{1},u_{3}),(u_{2},u_{1}),(u_{2},u_{3}),(u_{4},u_{2})$. Let $D^{\prime}=D-\{u_{1},u_{2}\}$. Then

[TABLE]

In particular, $D$ is an AT-orientation if and only if $D^{\prime}$ is an AT-orientation.

Proof.

Each Eulerian sub-digraph of $D^{\prime}$ is an Eulerian sub-digraph of $D$ (with $u_{1},u_{2}$ being isolated vertices). On the other hand, if $F$ is an Eulerian sub-digraph of $D$ but $F-\{u_{1},u_{2}\}$ is not an Eulerian sub-digraph of $D^{\prime}$, then $(u_{4},u_{2})\in F$, and exactly one of $P_{1}=(u_{2},u_{3})$ and $P_{2}=(u_{2},u_{1},u_{3})$ is contained in $F$. For $i\in[2]$, let $\mathcal{E}_{i}$ be the Eulerian sub-digraphs of $D$ containing $P_{i}$. If $F\in\mathcal{E}_{i}$, then $F^{\prime}=(F-P_{i})\cup P_{3-i}\in\mathcal{E}_{3-i}$, and $F$ and $F^{\prime}$ have different parities. So the Eulerian sub-digraphs of $D$ that are not Eulerian sub-digraphs of $D^{\prime}$ contributes [math] to the difference ${\rm diff}(D)$ of $D$. Hence ${\rm diff}(D)={\rm diff}(D^{\prime}).$ ∎

Lemma 2.3**.**

Assume $G$ is a $K_{4}$-minor-free graph and $X$ is a set of three vertices of $G$. If there is a proper $3$-coloring $\phi$ of $G$ such that $|\phi(X)|=2$, then $X$ is feasible.

Proof.

Since $|X|=3$ and $|\phi(X)|=2$, by 1.4, $X$ is not connected by a chain of diamonds. As $|\phi(X)|=2$, $X$ is feasible. ∎

Corollary 2.4**.**

Assume $G$ is a $K_{4}$-minor-free graph, $xx^{\prime}$ is an edge of $G$, and $y,z\not\in\{x,x^{\prime}\}$. Then at least one of the sets $\{x,y,z\}$ and $\{x^{\prime},y,z\}$ is feasible.

Proof.

Assume $xx^{\prime}$ is an edge of $G$ and $\phi$ is a proper 3-coloring of $G$. Since $\phi(x)\neq\phi(x^{\prime})$, for any vertices $y,z\not\in\{x,x^{\prime}\}$, at least one of $\phi(\{x,y,z\})$ and $\phi(\{x^{\prime},y,z\})$ has size 2. Hence, by Lemma 2.3, at least one of $\{x,y,z\}$ and $\{x^{\prime},y,z\}$ is feasible. ∎

Observation 2.5**.**

Assume $G$ is a connected $K_{4}$-minor-free graph with minimum degree $2$ that is not a cycle. It is well-known [7] that $G$ has a plane embedding where two faces of $G$ each has an incident $2$-vertex. In particular, $G$ has at least two non-adjacent $2$-vertices.

Definition 2.6**.**

Assume $G$ is a connected $K_{4}$-minor-free graph with minimum degree $2$ that is not a cycle. Assume $v$ is a $2$-vertex on a triangle $[vu_{1}u_{2}]$, and if $u_{i}$ is a $3$-vertex, then let $w_{i}\in N_{G}(u_{i})\setminus\{v,u_{3-i}\}$. We say $v$ is genuine if (1) or (2) holds:

1. (1)

$d_{G}(u_{i})=3$ and $w_{i}u_{3-i}$ is an edge of $G$ for some $i\in[2]$. 2. (2)

$d_{G}(u_{1})=d_{G}(u_{2})=3$ and $w_{1}w_{2}$ is an edge of $G$.

Figure 3 is an illustration of genuine vertices, where a vertex represented by a square in the figure indicates that its degree is unspecified.

Lemma 2.7**.**

Assume $G$ is a connected $K_{4}$-minor-free graph with minimum degree $2$ that is not a cycle, and $G$ has at most three $2$-vertices. If $G$ has only two $2$-vertices, then both are genuine $2$-vertices. If $G$ has exactly three $2$-vertices, then at least one of them is a genuine $2$-vertex.

Proof.

For each non-genuine $2$-vertex $v$ whose neighbors $u_{1},u_{2}$ are $3^{+}$-vertices, if $u_{1}u_{2}$ is not an edge of $G$, then contract the edge $vu_{1}$. If $u_{1}u_{2}$ is an edge of $G$, then we do the following operation:

If both $u_{1},u_{2}$ are $4^{+}$-vertices, then delete $v$. 2. 2.

If both $u_{1},u_{2}$ are $3$-vertices, where $w_{i}\in N_{G}(u_{i})\setminus\{v,u_{3-i}\}$ for $i\in[2]$, $w_{1}\neq w_{2}$, and $w_{1}w_{2}$ is not an edge of $G$, then contract all the edges $vu_{1},vu_{2},u_{1}u_{2},u_{1}w_{1}$. 3. 3.

If $u_{1}$ is a $3$-vertex and $u_{2}$ is a $4^{+}$-vertex, where $w_{1}\in N_{G}(u_{1})\setminus\{v,u_{2}\}$, and $w_{1}u_{2}$ is not an edge of $G$, then contract all the edges $vu_{1},vu_{2},u_{1}u_{2}$.

We denote by $G^{\prime}$ the resulting graph. It follows from the construction that each non-genuine $2$-vertex of $G$ with two $3^{+}$-neighbors is not a vertex of $G^{\prime}$, and no new $2$-vertices are created. In other words, every vertex of $G^{\prime}$ has degree at least $3$, except genuine $2$-vertices of $G$ or $2$-vertices of $G$ with a $2$-neighbor in $G$. Note that $G^{\prime}$ is also a connected $K_{4}$-minor-free graph with minimum degree at least $2$ that is not a cycle, so that $G^{\prime}$ also has at least two non-adjacent $2$-vertices by 2.5.

If $G$ has only two $2$-vertices, which are non-adjacent by 2.5, and at least one them is not genuine, then $G^{\prime}$ has at most one $2$-vertex, which is a contradiction. Thus, if $G$ has only two $2$-vertices, then both are genuine $2$-vertices.

If $G$ has exactly three $2$-vertices, and none of them are genuine $2$-vertices, then every $2$-vertex in $G^{\prime}$ is a $2$-vertex that is adjacent to a $2$-vertex in $G$. Since $G$ has exactly three $2$-vertices, every $2$-vertex in $G^{\prime}$ should be on the same face of $G^{\prime}$, which is a contradiction to 2.5. Thus, if $G$ has exactly three $2$-vertices, then at least one of them is a genuine $2$-vertex. ∎

A Gallai tree is a graph in which every block is a complete graph or an odd cycle. The following theorem is known as the degree-AT theorem.

Theorem 2.8** (The degree-AT theorem[3]).**

Let $G$ be a connected graph. If $G$ is not a Gallai tree, then $G$ has an AT orientation $D$ such that $d_{D}^{-}(v)\geq 1$ for each $v\in V(G)$.

3 Proof of Theorem 1.5

Assume $G$ is a $K_{4}$-minor-free graph and $x,y$ are distinct vertices of $G$. We prove by induction on the number of vertices of $G$ that $G$ is $(2,2)$-AT extendable with respect to $(x,y)$, and if $\{x,y\}$ is not connected by a chain of diamonds, then $G$ is $(2,1)$-AT extendable with respect to $(x,y)$.

By induction, we may assume that $G$ is connected. If $G$ is a subgraph of a cycle, then it is easily checked that for any distinct vertices $x,y$ of $G$, $G$ is $(2,1)$-AT extendable with respect to $(x,y)$. Thus, assume $G$ is not a subgraph of a cycle, and this implies that $G$ has at least four vertices.

If $d_{G}(v)\leq 2$ for some vertex $v\not\in\{x,y\}$, then by induction $G^{\prime}=G-v$ has a $(2,2)$-AT orientation (if $\{x,y\}$ is not connected by a chain of diamonds, then $(2,1)$-AT orientation) $D^{\prime}$ with respect to $(x,y)$. By orienting the edges incident with $v$ as out-arcs of $v$, we obtain a $(2,2)$-AT orientation (if $\{x,y\}$ is not connected by a chain of diamonds, then $(2,1)$-AT orientation) of $G$ with respect to $(x,y)$. Thus, we may assume that all vertices other than $x,y$ are $3^{+}$-vertices.

If $d_{G}(x)=1$, then $G-x$ has a $(2,1)$-AT orientation with respect to $(z,y)$, where $z$ is a vertex of $G-x$ such that $\{z,y\}$ is not connected by a chain of diamonds. This can be extended to a $(2,1)$-AT orientation of $G$ with respect to $(x,y)$ by orienting the edge incident with $x$ as an out-arc of $x$. Thus, we may assume that $x$ is a $2^{+}$-vertex.

Suppose $d_{G}(y)=1$, and $N_{G}(y)=\{z\}$. If $x=z$, then $G-y$ has a $(2,1)$-AT orientation with respect to $(w,x)$, where $w$ is a vertex of $G-y$ such that $\{w,x\}$ is not connected by a chain of diamonds. This can be extended to a $(2,1)$-AT orientation of $G$ with respect to $(x,y)$ by orienting the edge incident with $y$ as an in-arc of $y$. If $x\neq z$, then $G-y$ has a $(2,2)$-AT orientation with respect to $(z,x)$, which can be extended to a $(2,1)$-AT orientation of $G$ with respect to $(x,y)$ by orienting the edge incident with $y$ as an in-arc of $y$. Thus, we may assume that $y$ is a $2^{+}$-vertex.

Thus $G$ has minimum degree 2, and all vertices other than $x,y$ are $3^{+}$-vertices. So both $x,y$ are genuine $2$-vertices of $G$ by Lemma 2.7.

As $x$ is a genuine $2$-vertex of $G$, we may assume that $[xu_{1}u_{2}]$ is a triangle, $d_{G}(u_{1})=3$ and $N_{G}(u_{1})=\{x,u_{2},w_{1}\}$.

Case 1. $u_{2}w_{1}$ is an edge of $G$.

As $[xu_{1}u_{2}]$ is a triangle, $\{x,w_{1}\}$ is connected by a chain of diamonds. If $w_{1}\neq y$, then by induction $G^{\prime}=G-\{x,u_{1}\}$ has a $(2,2)$-AT orientation $D^{\prime}$ with respect to $(w_{1},y)$. Add the arcs $(w_{1},u_{1}),(u_{1},x),(x,u_{2}),(u_{1},u_{2})$ to $D^{\prime}$ to obtain an orientation $D$ of $G$. By Lemma 2.2, ${\rm diff}(D)={\rm diff}(D^{\prime})$. Hence $D$ is a $(2,2)$-AT orientation of $G$ with respect to $(x,y)$. If $\{x,y\}$ is not connected by a chain of diamonds, then $\{w_{1},y\}$ is not connected by a chain of diamonds. Hence we may assume that $D^{\prime}$ is a $(2,1)$-AT orientation of $G^{\prime}$ with respect to $(w_{1},y)$, and therefore $D$ is a $(2,1)$-AT orientation of $G$ with respect to $(x,y)$.

If $w_{1}=y$, then $\{u_{2},y\}$ is not connected by a chain of diamonds. So by induction, $G^{\prime}=G-\{x,u_{1}\}$ has a $(2,1)$-AT orientation $D^{\prime}$ with respect to $(u_{2},y)$. Add the arcs $(y,u_{1})$, $(u_{1},x)$, $(x,u_{2})$, $(u_{1},u_{2})$ to $D^{\prime}$ to obtain a $(2,2)$-AT orientation $D$ of $G$ with respect to $(x,y)$.

Case 2. $u_{2}w_{1}$ is not an edge of $G$.

By the definition of a genuine $2$-vertex, $d_{G}(u_{2})=3$, $N_{G}(u_{2})=\{x,u_{1},w_{2}\}$, and $w_{1}w_{2}$ is an edge of $G$. Then either $\{w_{1},y\}$ or $\{w_{2},y\}$ is not connected by a chain of diamonds (note that if $y=w_{2}$, then $\{w_{1},y\}$ is not connected by a chain of diamonds). By symmetry, we may assume that $\{w_{1},y\}$ is not connected by a chain of diamonds. Then by induction $G^{\prime}=G-\{x,u_{1}\}$ has a $(2,1)$-AT orientation $D^{\prime}$ with respect to $(w_{1},y)$. Add the arcs $(w_{1},u_{1}),(u_{1},x),(x,u_{2}),(u_{1},u_{2})$ to $D^{\prime}$. By Lemma 2.2, the resulting orientation is a $(2,1)$-AT orientation of $G$ with respect to $(x,y)$. This completes the proof of Theorem 1.5.

4 Proof of Theorem 1.8

Assume $G$ is a $K_{4}$-minor-free graph and $x,y,z\in V(G)$ are distinct vertices. Suppose to the contrary that $\{x,y,z\}$ is a feasible set, but $G$ is not $(2,2,2)$-AT extendable with respect to $(x,y,z)$. Let $G$ be such a graph with the minimum number of vertices. Then $G$ is connected, and for the same reason as in the proof of Theorem 1.5, $G$ is not a subgraph of a cycle, and $d_{G}(x),d_{G}(y),d_{G}(z)\geq 2$, and $d_{G}(v)\geq 3$ for each $v\in V(G)\setminus\{x,y,z\}$.

Since $G$ is not a subgraph of a cycle, $G$ has at least four vertices. By Lemma 2.7, we may assume $x$ is a genuine $2$-vertex of $G$, and $[xu_{1}u_{2}]$ is a triangle, $d_{G}(u_{1})=3$ and $N_{G}(u_{1})=\{x,u_{2},w_{1}\}$.

Case 1. $u_{2}w_{1}$ is an edge of $G$.

Then $\{x,w_{1}\}$ is connected by a chain of diamonds so that $\{w_{1},y,z\}$ is feasible if $w_{1}\not\in\{y,z\}$.

Suppose $u_{1}\in\{y,z\}$. Say $u_{1}=z$. By Lemma 2.7, $y$ is a genuine $2$-vertex of $G$. If $y\neq w_{1}$, then we may switch $x$ and $y$ so that $u_{1}\neq z$. Note that after switching, it is possible that $w_{1}=z$. If $w_{1}=z$, then since $\{y,z\}$ is not connected by a chain of diamonds, by Theorem 1.5, $G^{\prime}=G=\{x,u_{1}\}$ has a $(2,1)$-AT orientation $D^{\prime}$ with respect to $(y,z)$. By Lemma 2.2, $D^{\prime}$ can be extended to a $(2,2,2)$-AT orientation $D$ of $G$, by adding the arcs $(z,u_{1}),(u_{1},x),(x,u_{2}),(u_{1},u_{2})$ to $D^{\prime}$, which is a contradiction. Thus, $w_{1}\neq z$. In this case, $\{w_{1},y,z\}$ is feasible in $G^{\prime}=G-\{x,u_{1}\}$, and by minimality, $G^{\prime}$ has a $(2,2,2)$-AT orientation $D^{\prime}$ with respect to $(w_{1},y,z)$. Add the arcs $(w_{1},u_{1})$, $(u_{1},x)$, $(x,u_{2})$, $(u_{1},u_{2})$ to obtain an orientation $D$ of $G$. By Lemma 2.2, ${\rm diff}(D)={\rm diff}(D^{\prime})$. Hence $D$ is a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$.

If $y=w_{1}$, then by Theorem 1.5, $G^{\prime}=G-\{x,y,z\}$ has a $(1)$-AT orientation $D^{\prime}$ with respect to $(u_{2})$. Add the arcs $(u_{2},y)$, $(y,z)$, $(z,u_{2})$, $(u_{2},x)$, $(x,z)$ to obtain an orientation $D$ of $G$. Let $G^{\prime\prime}$ be a subgraph of $G$ induced by $\{x,y,z,u_{2}\}$, and $D^{\prime\prime}$ be an orientation of $G^{\prime\prime}$ obtained by restricting $D$ onto $G^{\prime\prime}$. By Lemma 2.1, ${\rm diff}(D)={\rm diff}(D^{\prime})\times{\rm diff}(D^{\prime\prime})\neq 0$. Hence $D$ is a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$.

Suppose $u_{1}\not\in\{y,z\}$. If $w_{1}\not\in\{y,z\}$, then $\{w_{1},y,z\}$ is feasible in $G^{\prime}=G-\{x,u_{1}\}$, and by minimality, $G^{\prime}$ has a $(2,2,2)$-AT orientation $D^{\prime}$ with respect to $(w_{1},y,z)$. Add the arcs $(w_{1},u_{1})$, $(u_{1},x)$, $(x,u_{2})$, $(u_{1},u_{2})$ to obtain an orientation $D$ of $G$. By Lemma 2.2, ${\rm diff}(D)={\rm diff}(D^{\prime})$. Hence $D$ is a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$.

If $w_{1}\in\{y,z\}$, say $w_{1}=y$, then $\{y,z\}$ is not connected by a chain of diamonds. By Theorem 1.5, $G^{\prime}=G-\{x,u_{1}\}$ has a $(2,1)$-AT orientation $D^{\prime}$ with respect to $(z,y)$. Add the arcs $(y,u_{1}),(u_{1},x),(x,u_{2}),(u_{1},u_{2})$ to $D^{\prime}$ to obtain $D$. By Lemma 2.2, $D$ is a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$.

Case 2. $u_{2}w_{1}$ is not an edge of $G$.

Then $d_{G}(u_{2})=3$, $N_{G}(u_{2})=\{x,u_{1},w_{2}\}$ and $w_{1}w_{2}$ is an edge of $G$. If $\{u_{1},u_{2}\}\cap\{y,z\}\neq\emptyset$, say $u_{1}=z$, then by Lemma 2.7, $y$ is a genuine $2$-vertex, and we can switch $x$ and $y$ so that $\{u_{1},u_{2}\}\cap\{y,z\}=\emptyset$. Thus, we may assume that $\{u_{1},u_{2}\}\cap\{y,z\}=\emptyset$.

If $\{y,z\}=\{w_{1},w_{2}\}$, say $y=w_{1},z=w_{2}$, then by Theorem 1.5, $G^{\prime}=G-\{x,u_{1},u_{2}\}$ has a $(2,1)$-AT orientation $D^{\prime}$ with respect to $(z,y)$. Add the arcs $(y,u_{1})$, $(u_{1},x)$, $(x,u_{2})$, $(u_{1},u_{2})$, $(u_{2},z)$ to $D^{\prime}$ to obtain a $(2,2,2)$-AT orientation $D$ of $G$ with respect to $(x,y,z)$.

If $y,z\notin\{w_{1},w_{2}\}$, then by Corollary 2.4, $\{w_{1},y,z\}$ or $\{w_{2},y,z\}$ is feasible. By symmetry, we may assume that $\{w_{1},y,z\}$ is feasible. Then by minimality $G^{\prime}=G-\{x,u_{1},u_{2}\}$ has a $(2,2,2)$-AT orientation $D^{\prime}$ with respect to $(w_{1},y,z)$. By adding the arcs $(w_{1},u_{1})$, $(u_{1},x)$, $(x,u_{2})$, $(u_{1},u_{2})$, $(u_{2},w_{2})$, we obtain a $(2,2,2)$-AT orientation $D$ of $G$ with respect to $(x,y,z)$.

Assume $|\{y,z\}\cap\{w_{1},w_{2}\}|=1$, say $y=w_{1}$ and $z\notin\{w_{1},w_{2}\}$. If $\{y,z\}$ is not connected by a chain of diamonds, then by Theorem 1.5, $G^{\prime}=G-\{x,u_{1},u_{2}\}$ has a $(2,1)$-AT orientation $D^{\prime}$ with respect to $(z,y)$. Add the arcs $(y,u_{1}),(u_{1},x),(x,u_{2}),(u_{1},u_{2}),(u_{2},w_{2})$ to $D^{\prime}$ to obtain a $(2,2,2)$-AT orientation $D$ of $G$ with respect to $(x,y,z)$.

If $\{y,z\}$ is connected by a chain of diamonds, then for any proper 3-coloring $\phi$ of $G^{\prime}=G-\{x,u_{1},u_{2}\}$, $\phi(y)=\phi(z)$ and hence $|\{\phi(w_{2}),\phi(y),\phi(z)\}|=2$. So $\{w_{2},y,z\}$ is feasible, and by minimality, $G^{\prime}$ has a $(2,2,2)$-AT orientation $D^{\prime}$ with respect to $(w_{2},y,z)$. Add the arcs $(w_{2},u_{2}),(u_{2},x),(x,u_{1}),(u_{2},u_{1}),(u_{1},y)$ to $D^{\prime}$. By Lemma 2.2, the resulting orientation $D$ is a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$. This completes the proof of Theorem 1.8.

5 Proof of Theorem 1.12

Suppose to the contrary that there is a graph $G$ with $\operatorname{mad}(G)<\frac{14}{5}$, and $G$ is not $(2,2,2)$-AT extendable with respect to a non-blocked triple $(x,y,z)$. Let $G$ be a minimal graph with this property with respect to $|V(G)|$. By Corollary 1.9, $G$ has a $K_{4}$-minor.

Claim 5.1**.**

Every vertex $v\in V(G)\setminus\{x,y,z\}$ is a $3^{+}$-vertex in $G$, and $x,y,z$ are $2^{+}$-vertices in $G$.

Proof.

Suppose to the contrary that $v$ is a $2^{-}$-vertex in $V(G)\setminus\{x,y,z\}$. By minimality of $G$, $G-v$ has a $(2,2,2)$-AT orientation with respect to $(x,y,z)$. By orienting the edges incident with $v$ as out-arcs of $v$, we obtain a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$, which is a contradiction. Thus, every vertex in $V(G)\setminus\{x,y,z\}$ is a $3^{+}$-vertex in $G$.

Suppose to the contrary that $v$ is a $1^{-}$-vertex in $\{x,y,z\}$. Without loss of generality, let $v=x$. Let $x^{\prime}$ be a vertex such that $\{x^{\prime},y,z\}$ is non-blocked in $G^{\prime}=G-x$. By minimality of $G$, $G^{\prime}$ has a $(2,2,2)$-AT orientation with respect to $(x^{\prime},y,z)$. By orienting the edge incident with $x$ as an out-arc of $x$, we obtain a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$, which is a contradiction. Thus, $x,y,z$ are $2^{+}$-vertices in $G$. ∎

Let $n_{i}$ and $n_{i}^{+}$ be the number of $i$-vertices and $i^{+}$-vertices, respectively, in $G$. By Claim 5.1, $n_{0}=n_{1}=0$ and so

[TABLE]

Let B=\{w\in\{x,y,z\}:w\text{ is a 2-vertex}\}. By Claim 5.1,

[TABLE]

Thus $|B|\geq 2$ and $n_{5}^{+}=0$. Then the vertices with odd degree are all $3$-vertices, which implies that $n_{3}$ is even and so

[TABLE]

This also implies that $n_{4}\leq 1$.

In the following, we will find an orientation $D$ of $G$ such that $D$ is a $(2,2,2)$-AT orientation of $G$ with respect to $(x,y,z)$, that is, $\Delta^{+}(D)\leq 2$, $x,y,z$ have at most one out-arc in $D$, and $\operatorname{diff}(D)\neq 0$. Let $H^{*}$ be the graph obtained from $G$ by contracting an edge incident with a 2-vertex one by one. Note that $H^{*}$ may have multiple edges or loops. Since $G$ has a $K_{4}$-minor, $H^{*}$ also has a $K_{4}$-minor, and therefore $|V(H^{*})|\geq 4$. Given an orientation $D^{*}$ of $H^{*}$, obtaining an orientation $D$ of $G$ by the following is called a recovering process: For $(u,w)\in A(D^{*})$,

1. (i)

if $u\neq w$, then an edge $uw$ of $H^{*}$ corresponds to a path $v_{1}\ldots v_{k}$ in $G$ with internal $2$-vertices where $v_{1}=u$ and $v_{k}=w$ and so let $(v_{i},v_{i+1})\in A(D)$ for all $i\in[k-1]$; 2. (ii)

if $u=w$, then the loop $uw$ of $H^{*}$ corresponds to a cycle $v_{1}\ldots v_{k}$ with internal $2$-vertices in $G$ where $v_{1}=v_{k}=u$, then let $(u,v_{k-1})\in A(D)$ and let $(v_{i},v_{i+1})\in A(D)$ for all $i\in[k-2]$.

Case 1. $|B|=3$, that is, $x,y,z$ are all $2$-vertices.

If all vertices in $V(G)\setminus\{x,y,z\}$ are $3$-vertices, then since $G$ is not $(2,2,2)$-AT extendable with respect to $(x,y,z)$, by the degree-AT theorem (Theorem 2.8), $G$ is a Gallai tree such that every block must be an odd cycle or a $K_{2}$. This is a contradiction to the fact that $G$ has a $K_{4}$-minor. Thus $G$ has a unique $4$-vertex. By (5.1), $n_{3}\leq 4$. Since $|V(H^{*})|\geq 4$, $n_{3}=4$, and so $|V(G)|=8$. Then $G$ is a graph with degree sequence $(4,3,3,3,3,2,2,2)$, and $H^{*}$ is a graph with degree sequence $(4,3,3,3,3)$. Since $H^{*}$ has a $K_{4}$-minor, there are two vertices $v_{1}$ and $v_{2}$ of $H^{*}$ such that $V(H^{*})\setminus\{v_{1},v_{2}\}$ is a triangle and each vertex in $V(H^{*})\setminus\{v_{1},v_{2}\}$ has a neighbor in $\{v_{1},v_{2}\}$. Let $V(H^{*})\setminus\{v_{1},v_{2}\}=\{v_{3},v_{4},v_{5}\}$. By the degree constraint, a loop or a multiple edge is incident only with $v_{1}$ or $v_{2}$ if it exists. There are three possible graphs for $H^{*}$, and for each case, we give an orientation $D^{*}$ of $H^{*}$ as in Figure 4.

Let $D$ be an orientation of $G$ obtained from $D^{*}$ by the recovering process. Note that $D^{*}-v_{2}v_{5}$ is acyclic, any nonempty Eulerian sub-digraph of $D^{*}$ contains the arc $v_{2}v_{5}$, and so there are exactly two nonempty Eulerian sub-digraphs in $D^{*}$. Thus $D$ also has an odd number of Eulerian sub-digraphs and so $\operatorname{diff}(D)\neq 0$.

Case 2. $|B|=2$

We may assume that $x,y$ are $2$-vertices, and $z$ is a $3^{+}$-vertex in $G$. Since $n_{3}+n_{4}\geq 3$, it follows from (5.1) that $n_{4}=0$ and $n_{3}\in\{4,6\}$. Thus $|V(G)|\in\{6,8\}$.

Case 2-1. Suppose that $n_{3}=6$. Then $G$ is a graph with degree sequence $(3,3,3,3,3,3,2,2)$, and $H^{*}$ is a graph with degree sequence $(3,3,3,3,3,3)$. If $H^{*}$ has a loop, then $G$ has a 3-cycle $[u_{1}u_{2}u_{3}]$ such that $u_{1},u_{2}$ are 2-vertices and $u_{3}$ is a 3-vertex in $G$. A subgraph of $G$ induced by $V(G)\setminus\{u_{1},u_{2},u_{3}\}$ has average degree $\frac{14}{5}$. Since $\operatorname{mad}(G)<\frac{14}{5}$, $H^{*}$ has no loop. Since $H^{*}$ has a $K_{4}$-minor, $H^{*}$ is one of graphs in Figure 5: If $H^{*}$ has a multiple edge, then $H^{*}$ is (1), and if $H^{*}$ is simple, then $H^{*}$ is a $2$-connected cubic graph and so $H^{*}$ is (2) or (3).

By symmetry, we may assume that for (1), $z\in\{v_{1},v_{5},v_{6}\}$, and for (2) or (3), $z=v_{1}$. Let $D$ be an orientation of $G$ obtained from the orientation $D^{*}$ of $H^{*}$ in Figure 5 by the recovering process. In any case, $D^{*}-v_{3}v_{6}$ is acyclic and so any nonempty Eulerian sub-digraph of $D^{*}$ contains the arc $v_{3}v_{6}$. Then $D^{*}$ has exactly five Eulerian sub-digraphs for (1), and has exactly three Eulerian sub-digraphs for each of (2) and (3). Thus $D$ also has an odd number of Eulerian sub-digraphs and so $\operatorname{diff}(D)\neq 0$.

Case 2-2. Suppose that $n_{3}=4$. Then $G$ is a graph with degree sequence $(3,3,3,3,2,2)$, and $H^{*}$ must be $K_{4}$. Let $V(H^{*})=\{v_{1},v_{2},v_{3},v_{4}\}$. Since one $3$-vertex in $G$ must be $z$, we assume that $v_{1}=z$. Note that by the degree constraint, subdividing edges of $H^{*}$ twice makes $G$, and so $H^{*}$ has at most two edges that are not edges of $G$.

Suppose that $v_{2}v_{3},v_{3}v_{4},v_{2}v_{4}$ are edges of $G$. Since $x,y,z$ is non-blocked, two edges of $H^{*}$ incident with $v_{1}$ are not edges of $G$. We may assume that $v_{1}v_{3}$ and $v_{1}v_{4}$ are not edges of $G$. Let $D$ be the orientation of $G$ obtained from the orientation $D^{*}$ depicted in (1) of Figure 6 by the recovering process. Then $D$ has exactly one odd Eulerian sub-digraph $v_{2}v_{4}v_{3}v_{2}$, and three even Eulerian sub-digraphs. Thus $\operatorname{diff}(D)\neq 0$.

Suppose that one of $v_{2}v_{3},v_{3}v_{4},v_{2}v_{4}$ is not an edge of $G$, say $v_{2}v_{4}$ is not an edge. Let $D_{0}$ be the orientation depicted in (2) of Figure 6, where the double line shows a schematic representation of direction to define an orientation $D$ of $G$ using $D_{0}$. The path $v_{2}u_{1}\ldots u_{t}v_{4}$ of $G$ corresponding to $v_{2}v_{4}$ of $H^{*}$ is oriented so that $(v_{2},u_{1})$, $(v_{4},u_{t})$ are arcs of $D$ and $u_{1}\ldots u_{t}$ is a directed path. For the other edge of $H^{*}$ not in $G$, we naturally extend the orientation $D_{0}$ so that an arc of $D_{0}$ is a directed path in $D$. The resulting orientation $D$ of $G$ has three Eulerian sub-digraphs. Thus $\operatorname{diff}(D)\neq 0$. This completes the proof.

Acknowledgements

Eun-Kyung Cho was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Education (No. RS-2023-00244543 and No. RS-2023-00211670). Ilkyoo Choi was supported in part by the Hankuk University of Foreign Studies Research Fund and by the Institute for Basic Science (IBS-R029-C1). Boram Park was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2025-00523206), and supported by the New Faculty Startup Fund from Seoul National University. Xuding Zhu was supported by the National Natural Science Foundation of China, Grant numbers: NSFC 12371359, U20A2068.