Nice vertices in cubic graphs

TL;DR

This paper investigates the number of nice vertices in matching covered cubic graphs, establishing lower bounds for non-bipartite and bipartite cases, and characterizing extremal graphs with minimal nice vertices or pairs.

Contribution

It provides new lower bounds on the number of nice vertices in 2- and 3-connected cubic graphs and characterizes extremal graphs, including the unique extremal bipartite case.

Findings

Non-bipartite 2-connected cubic graphs have at least 4 nice vertices.

3-connected non-bipartite cubic graphs (not K4) have at least 6 nice vertices.

Connected bipartite cubic graphs have at least 9 nice pairs, with K_{3,3} as the extremal case.

Abstract

A subgraph $G'$ of a graph $G$ is nice if $G-V(G')$ has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex $u$ of a cubic graph is nice if $u$ and its neighbors induce a nice subgraph. D. Kr\'{a}l et al. (2010) [9] showed that each vertex of a cubic brick is nice. It is natural to ask how many nice vertices a matching covered cubic graph has. In this paper, using some basic results of matching covered graphs, we prove that if a non-bipartite cubic graph $G$ is 2-connected, then $G$ has at least 4 nice vertices; if $G$ is 3-connected and $G\neq K_4$, then $G$ has at least 6 nice vertices. We also determine all the corresponding extremal graphs. For a cubic bipartite graph $G$ with bipartition $(A,B)$, a pair of vertices $a\in A$ and $b\in B$ is called a nice pair if $a$ and $b$ together with their neighbors induce a nice subgraph. We show that a connected cubic bipartite graph $G$ is a brace if and only if each pair of vertices in distinct color classes is a nice pair. In general, we prove that $G$ has at least 9 nice pairs of vertices and $K_{3,3}$ is the only extremal graph.

Full text

Nice vertices in cubic graphs${}^{\text{\char 73}\ddagger}$

Wuxian Chen1, Fuliang Lu2, Heping Zhang1†

1School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, PR China

2School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, 363000, PR China

††${}^{\text{\char 73}}$ This work is supported by NSFC (Grant No. 12271229 and 12271235) and NSF of Fujian Province (Grant No. 2021J06029).††† Corresponding author. E-mail addresses: chenwx21@lzu.edu.cn (W. Chen), flianglu@163.com (F. Lu) and zhanghp@lzu.edu.cn (H. Zhang).††‡ Published in [Discrete Math. 348 (2025) 114553]. This version fixes an error in Lemma 3.2, and thus corrects the proofs of Lemmas 3.3 and 3.4. See the appendix for details.

Abstract: A subgraph $G^{\prime}$ of a graph $G$ is nice if $G-V(G^{\prime})$ has a perfect matching. Nice subgraphs play a vital role in the theory of ear decomposition and matching minors of matching covered graphs. A vertex $u$ of a cubic graph is nice if $u$ and its neighbors induce a nice subgraph. D. Král et al. (2010) [9] showed that each vertex of a cubic brick is nice. It is natural to ask how many nice vertices a matching covered cubic graph has. In this paper, using some basic results of matching covered graphs, we prove that if a non-bipartite cubic graph $G$ is 2-connected, then $G$ has at least 4 nice vertices; if $G$ is 3-connected and $G\neq K_{4}$, then $G$ has at least 6 nice vertices. We also determine all the corresponding extremal graphs.

For a cubic bipartite graph $G$ with bipartition $(A,B)$, a pair of vertices $a\in A$ and $b\in B$ is called a nice pair if $a$ and $b$ together with their neighbors induce a nice subgraph. We show that a connected cubic bipartite graph $G$ is a brace if and only if each pair of vertices in distinct color classes is a nice pair. In general, we prove that $G$ has at least 9 nice pairs of vertices and $K_{3,3}$ is the only extremal graph.

Keywords: nice vertex; nice pair; cubic graph; matching covered graph; perfect matching

AMS subject classification: 05C70, 05C75

1 Introduction

Graphs considered in this paper are connected, finite and simple unless specified. A graph is cubic if each vertex has degree 3. Cubic graphs have been studied extensively since various open problems and conjectures can be reduced to cubic graphs. A matching in a graph $G$ is a set of independent edges. A perfect matching of $G$ is a matching covering all vertices of $G$. In 1891, Petersen [19] showed a classical result that every 2-connected cubic graph has a perfect matching. Schönberger [22] established a stronger result that each edge of a 2-connected cubic graph lies in a perfect matching. This implies that every 2-connected cubic graph has at least three perfect matchings. Since then, extensive research has been devoted to the enumeration of perfect matchings of cubic graphs (see literature [9, 6, 7, 8]). Meanwhile, many structural properties of cubic graphs, especially fullerene graphs, have been obtained [4, 5, 15, 17].

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For a vertex subset $S$ of $G$, denote by $G-S$ the subgraph of $G$ obtained by deleting all the vertices in $S$ and their incident edges. A subgraph $G^{\prime}$ of $G$ is nice if $G-V(G^{\prime})$ has a perfect matching. Nice subgraphs are closely related to the theory of ear decomposition and matching minors of matching covered graphs (cf. [14, 13, 24]), which are also referred to as conformal subgraphs [2], central subgraphs [20] and well-fitted subgraphs [16]. A connected graph with at least two vertices is matching covered if each edge is contained in a perfect matching. So every 2-connected cubic graph is matching covered. Note that every matching covered graph on at least four vertices is 2-connected. Thus we have the following result.

Lemma 1.1**.**

A cubic graph is 2-connected if and only if it is matching covered.

Lovász and Plummer [14] showed that every matching covered graph on at least 6 vertices has a nice even cycle of length at least 6. Lovász [10] proved that every non-bipartite matching covered graph has an even subdivision of $K_{4}$ or $\overline{C_{6}}$ (the triangle prism) as a nice subgraph, where $K_{i}$ is a complete graph on $i$ vertices. In chemistry, nice subgraphs of a molecular graph $G$ represent addition patterns on $G$ such that the rest of $G$ still has a resonant structure. It has been proved that a fullerene graph (planar cubic graph with only pentagons and hexagons as faces) has some nice subgraphs, such as $K_{2},K_{1,3},P_{4}$ in [4], and $C_{6}$ in [25] and $C_{8}$ in [15], etc., where $P_{i}$ (resp. $C_{i}$) denotes a path (resp. cycle) on $i$ vertices and $K_{m,n}$ denotes a complete bipartite graph with bipartite sets having $m$ and $n$ vertices.

A spanning subgraph of a graph $G$ is a subgraph containing all vertices of $G$. Král et al. introduced the concept of trimatched vertices in cubic graphs in [9], where they proved that every 2-connected cubic graph on $n$ vertices has at least $\frac{n}{2}$ perfect matchings. A vertex $u$ of a cubic graph $G$ is trimatched if there exists a spanning subgraph $G^{\prime}$ of $G$ such that the degree of $u$ is 3 in $G^{\prime}$ and the degrees of the other vertices are 1 in $G^{\prime}$. That is, the subgraph $K_{1,3}$, which is induced by $u$ and its neighbors of $G^{\prime}$, is nice in $G$. Note that each $K_{1,3}$ subgraph of a cubic graph is nice if and only if the subgraph induced by $V(K_{1,3})$ is nice. From this point of view, if a vertex $u$ and its neighbors of a cubic graph induce a nice subgraph, then we call $u$ a nice vertex. Recall that a graph $G$ with at least one edge is bicritical if the removal of any two distinct vertices of $G$ yields a graph with a perfect matching. Clearly, a bicritical graph is matching covered. A 3-connected bicritical graph is called a brick. Král et. al. [9] showed that each vertex of a cubic brick is nice. In fact, their proof also implies the following more general result.

Lemma 1.2**.**

Each vertex of a cubic bicritical graph is nice.

A connected graph with at least 6 vertices and a perfect matching is 2-extendable if any two disjoint edges is contained in a perfect matching. A 2-extendable bipartite graph is called a brace. Using Lemma 1.2, Král et. al. showed that every nontrivial brick and brace decomposition of a 2-connected cubic graph $G$ contains a brace, which provides an ideal to prove the main theorem by using induction on the number of braces in nontrivial brick and brace decomposition of $G$. In this paper, we are interested in considering how many nice vertices a 2-connected cubic graph has. By using some basic results of matching covered graphs, we prove that if a non-bipartite cubic graph $G$ is 2-connected, then $G$ has at least 4 nice vertices; if $G$ is 3-connected and $G\neq K_{4}$, then $G$ has at least 6 nice vertices. We also determine all the corresponding extremal graphs (see Theorem 3.1).

Obviously, a cubic bipartite graph has no nice vertices. But we can consider a variant. Let $G(A,B)$ be a cubic bipartite graph with bipartition $(A,B)$. A pair of vertices $a\in A$ and $b\in B$ of $G(A,B)$ is a nice pair if $a$ and $b$ together with their neighbors induce a nice subgraph. Further, a pair of vertex subsets $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$ of $G(A,B)$ is a nice pair set if each pair of vertices $a\in A^{\prime}$ and $b\in B^{\prime}$ is nice. We show that a connected cubic bipartite graph $G(A,B)$ is a brace if and only if each pair of vertices $a\in A$ and $b\in B$ is nice. Then we prove that $G(A,B)$ has a nice pair set $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$ with $|A^{\prime}|\geq 3$ and $|B^{\prime}|\geq 3$ (see Theorem 4.1). As a corollary, we have that $G(A,B)$ has at least 9 nice pairs of vertices and $K_{3,3}$ is the only graph attaining the lower bound (see Corollary 4.2).

In Section 2, we present some notations and basic useful results of matching covered graphs. In Section 3, we prove Theorem 3.1 and determine all the extremal graphs by the splicing operation. In Section 4, we prove Theorem 4.1 and Corollary 4.2 and determine the corresponding extremal graphs. In the last section, we summarize our results and list some problems possibly for future research.

2 Preliminaries

In the section, we give some definitions and preliminary results. For a vertex $u$ of a graph $G$, a neighbor of $u$ is a vertex adjacent to $u$. Denote by $N_{G}(u)$ the set of neighbors of $u$ and $N_{G}[u]$ the closed neighborhood of $u$ in $G$, i.e., $N_{G}[u]:=N_{G}(u)\cup\{u\}$. The degree of $u$ in $G$, denoted by $d_{G}(u)$, is $|N_{G}(u)|$. An isolated vertex of $G$ is a vertex with degree 0. For a vertex subset $S$ of $G$, let $N_{G}(S)=\cup_{x\in S}N_{G}(x)$ and $G[S]$ be the subgraph of $G$ induced by $S$. A component $K$ of $G-S$ is odd if $|V(K)|$ is odd. Denote by $o(G-S)$ the number of odd components of $G-S$. The following is the well-known Tutte’s Theorem.

Theorem 2.1** ([23]).**

A graph $G$ has a perfect matching if and only if $o(G-S)\leq|S|$ for any $S\subseteq V(G)$.

Let $G$ be a graph with a perfect matching. A barrier of $G$ is a set $S\subseteq V(G)$ such that $o(G-S)=|S|$. Moreover, a barrier $S$ of $G$ is nontrivial if $|S|\geq 2$. We have the following basic property of matching covered graphs.

Lemma 2.2** ([3]).**

*Let $G$ be a matching covered graph. Then the following statements hold.

(i) $G$ is free of nontrivial barriers if and only if $G$ is bicritical,

(ii) For any barrier $S$ of $G$, $G-S$ has no even components,

(iii) For any barrier $S$ of $G$, $S$ is an independent set.*

Using Tutte’s Theorem, we give a necessary and sufficient condition under which a vertex of a 2-connected cubic graph is not nice as follows.

Theorem 2.3**.**

Let $G$ be a 2-connected cubic graph and $u$ be a vertex of $G$. Then $u$ is not nice in $G$ if and only if there exists a barrier $S$ of $G$ such that $u$ is an isolated vertex of $G-S$.

Proof.

Sufficiency. Since $u$ is an isolated vertex of $G-S$, $N_{G}(u)\subseteq S$. Set $S^{\prime}=S\setminus N_{G}(u)$. Then $o(G-N_{G}[u]-S^{\prime})=o(G-(S\cup\{u\}))=o(G-S)-1$. Since $S$ is a barrier and $d_{G}(u)=3$, $o(G-S)-1=|S|-1>|S|-3=|S^{\prime}|$. So $o(G-N_{G}[u]-S^{\prime})>|S^{\prime}|$. By Tutte’s Theorem, $G-N_{G}[u]$ has no perfect matching. So $u$ is not nice in $G$.

Necessity. Since $u$ is not nice, $G-N_{G}[u]$ has no perfect matching. By Tutte’s Theorem, there exists a set $S^{*}\subseteq V(G-N_{G}[u])$ such that $o(G-N_{G}[u]-S^{*})>|S^{*}|$. Then $o(G-N_{G}[u]-S^{*})\geq|S^{*}|+2$ by parity. Set $N_{G}(u)=\{u_{1},u_{2},u_{3}\}$. Since $G$ is matching covered by Lemma 1.1, $G-\{u,u_{1}\}$ has a perfect matching. By Tutte’s Theorem again, $o(G-\{u,u_{1}\}-(S^{*}\cup\{u_{2},u_{3}\}))\leq|S^{*}\cup\{u_{2},u_{3}\}|=|S^{*}|+2$. So $o(G-N_{G}[u]-S^{*})=|S^{*}|+2$ and $o(G-N_{G}(u)-S^{*})=o(G-N_{G}[u]-S^{*})+1=|S^{*}|+3=|S^{*}\cup N_{G}(u)|$. Thus $S:=N_{G}(u)\cup S^{*}$ is a barrier of $G$ such that $u$ is an isolated vertex of $G-S$. ∎

Let $X$ and $Y$ be two vertex subsets of a graph $G$ and $E_{G}(X,Y)$ (or simply $E(X,Y)$) the set of all edges of $G$ with one end in $X$ and the other in $Y$. For $\emptyset\neq X\subset V(G)$, the set $E(X,\overline{X})$ is called an edge cut of $G$, and is denoted by $\partial_{G}(X)$ (or simply by $\partial(X)$), where $\overline{X}:=V(G)\setminus X$. For a vertex $u$ of $G$, we denote $\partial{(\{u\})}$ by $\partial{(u)}$ for brevity. An edge cut $\partial(X)$ is trivial if either $|X|=1$ or $|\overline{X}|=1$. A k-cut is an edge cut with $k$ edges. For an edge cut $\partial(X)$ of $G$, denote by $G/(X\rightarrow x)$ (or simply $G/X$) the graph (possible with multiple edges) obtained from $G$ by contracting $X$ into a single vertex $x$. The two graphs $G/X$ and $G/\overline{X}$ are called the $\partial(X)$-contractions of $G$. We will use the following two facts related to edge cuts.

Lemma 2.4** ([1], Exercise 9.3.8).**

If $\partial{(X)}$ is a $k$-cut of a $k$-edge-connected graph, then the two $\partial(X)$-contractions are $k$-edge-connected.

Lemma 2.5** ([17]).**

In a 3-connected graph, every nontrivial 3-cut is a matching.

Let $G$ be a matching covered graph. An edge cut $C$ of $G$ is tight if $|C\cap M|=1$ for every perfect matching $M$ of $G$. It is clear that each trivial edge cut of $G$ is tight and the two $C$-contractions of $G$ are matching covered if $C$ is a tight cut of $G$.

Lemma 2.6** ([17]).**

Each tight cut of a 2-connected cubic graph is a 3-cut.

For a vertex subset $X$ of a bipartite graph $G(A,B)$, if $|X|$ is odd, then clearly one of the two sets $A\cap X$ and $B\cap X$ is larger than the other; we denote the larger and smaller sets by $X_{+}$ and $X_{-}$, respectively, and $\overline{X}_{+}$ and $\overline{X}_{-}$ are defined analogously for the vertex subset $\overline{X}$. The following is a characterization of tight cuts in bipartite matching covered graphs (see Fig. 1(a) for an example).

Lemma 2.7** ([11]).**

An edge cut $\partial{(X)}$ of a bipartite matching covered graph $G(A,B)$ is tight if and only if: (i) $|X|$ is odd and $|X_{+}|=|X_{-}|+1$, and (ii) $E(X_{-},\overline{X}_{-})=\emptyset$.

Lovász [11] showed the following important theorem that plays an essential role in our proof.

Theorem 2.8**.**

A matching covered graph has no nontrivial tight cuts if and only if it is either a brick (which is non-bipartite) or a brace.

Lemma 2.9** ([18]).**

A connected bipartite graph $G(A,B)$ with at least 6 vertices is a brace if and only if $G-\{a_{1},a_{2},b_{1},b_{2}\}$ has a perfect matching for any four distinct vertices $a_{1},a_{2}\in A$ and $b_{1},b_{2}\in B$.

Lemma 2.10** ([12]).**

Let $G$ be a bipartite matching covered graph that is not a brace. Then $G$ has a nontrivial tight cut $C$ such that one of the two $C$-contractions is a brace.

In the sequel, we show some results helpful to our proofs.

Lemma 2.11**.**

*Let $G$ be a 3-connected cubic graph with a nontrivial barrier $S$. Then the following statements hold.

(i) For any nontrivial component $K$ of $G-S$, $\partial{(V(K))}$ is a tight 3-cut and is a matching,

(ii) For any nontrivial component $K$ of $G-S$, each $\partial{(V(K))}$-contraction is a 3-connected simple cubic graph,

(iii) If $G$ is non-bipartite, then $G-S$ has a nontrivial non-bipartite component.*

Proof.

(i) Since $G$ is matching covered (Lemma 1.1), $K$ is odd by Lemma 2.2(ii). For any perfect matching $M$ of $G$, $|\partial{(V(K))}\cap M|=1$ as $o(G-S)=|S|$. So $\partial{(V(K))}$ is a tight cut. By Lemma 2.6, $\partial{(V(K))}$ is a 3-cut. Further, as $\partial{(V(K))}$ is nontrivial, $\partial{(V(K))}$ is a matching by Lemma 2.5, and so (i) holds.

(ii) Let $G^{\prime}$ be a $\partial{(V(K))}$-contraction. As $G$ is cubic, $G^{\prime}$ is also a simple cubic graph by (i). Since $|\partial{(V(K))}|=3$ by (i) and the fact that the edge connectivity of a cubic graph is equal to its vertex connectivity, $G^{\prime}$ is 3(-edge)-connected by Lemma 2.4, and so (ii) holds.

(iii) Otherwise, suppose that all nontrivial components of $G-S$ are bipartite graphs $H_{i}(A_{i},B_{i})$, $1\leq i\leq t$, with $|A_{i}|\geq|B_{i}|$, where $1\leq t\leq|S|$. Set $l_{i}=|E(A_{i},S)|$, $1\leq i\leq t$. By (i), $|\partial{(V(H_{i}))}|=3$ and so $0\leq l_{i}\leq 3$. Since $G$ is cubic, $|E(H_{i})|=3|A_{i}|-l_{i}=3|B_{i}|-(3-l_{i})$, which implies that $l_{i}=3$, $E(B_{i},S)=\emptyset$ and $|A_{i}|=|B_{i}|+1$. Then $(\cup_{i=1}^{t}B_{i}\cup S,\overline{\cup_{i=1}^{t}B_{i}\cup S})$ is a bipartition of $G$, contradicting that $G$ is non-bipartite. So (iii) holds. ∎

Lemma 2.12**.**

Let $G$ be a 2-connected cubic graph. Suppose that $\partial{(X)}$ is a nontrivial tight cut of $G$ such that $G/\overline{X}$ is simple. If $u\in X$ is nice in $G/\overline{X}$, then $u$ is nice in $G$.

Proof.

Set $G_{1}=G/(\overline{X}\rightarrow\overline{x})$ and $G_{2}=G/X$. Since $u$ is nice in $G_{1}$, $G_{1}-N_{G_{1}}[u]$ has a perfect matching, say $M_{1}$. As $G_{1}$ is simple and $u\neq\overline{x}$, $(M_{1}\cup\partial_{G_{1}}{(u)})\cap\partial_{G_{1}}{(\overline{x})}$ consists of exactly one edge, say $e_{1}$, which corresponds to the edge $e_{2}$ in $G_{2}$ and the edge $e$ in $G$. Since $G$ is matching covered by Lemma 1.1, $G_{2}$ is also matching covered. So by removing the two ends of $e_{2}$ from $G_{2}$, the left graph has a perfect matching, denoted by $M_{2}$. If $e_{1}\in\partial_{G_{1}}{(u)}$, then $M_{1}\cup M_{2}$ is a perfect matching of $G-N_{G}[u]$; if $e_{1}\notin\partial_{G_{1}}{(u)}$, then $(M_{1}\setminus\{e_{1}\})\cup M_{2}\cup\{e\}$ is a perfect matching of $G-N_{G}[u]$. Therefore, $u$ is nice in $G$. ∎

Lemma 2.12 can also be deduced by Lemma 21 (in which $G/\overline{X}$ and $G/X$ are both simple) in [9].

The union of two graphs $G_{1}$ and $G_{2}$ is the graph $G_{1}\cup G_{2}$ with vertex set $V(G_{1})\cup V(G_{2})$ and edge set $E(G_{1})\cup E(G_{2})$. The following result is a simple observation.

Lemma 2.13**.**

Let $G_{1}$ and $G_{2}$ be two vertex-disjoint bipartite graphs. If $u_{i1}$ and $u_{i2}$ are two vertices of $G_{i}$ in different color classes for $i=1,2$, then $(G_{1}\cup G_{2})+\{u_{11}u_{2j},u_{12}u_{2(3-j)}\}$ is a bipartite graph for $j=1,2$.

Lemma 2.14**.**

*Let $G$ be a 2-connected cubic graph. Assume that $\partial{(X)}=\{ab,cd\}$ is a 2-cut of $G$ such that $a,c\in X$. Then the following statements hold.

(i) $\partial{(X\setminus\{a\})}$ is a tight cut of $G$,

(ii) $G[X\setminus\{a,c\}]$ and $G[\overline{X}\setminus\{b,d\}]$ both have perfect matchings,

(iii) If $ac\notin E(G)$ and $u\in X$ is nice in $G[X]+ac$, then $u$ is nice in $G$,

(iv) If $G[X]$ is bipartite, then $a$ and $c$ are in distinct color classes of $G[X]$.*

Proof.

(i) Since $G$ is cubic, $2|E(G[X])|=\sum_{x\in X}d_{G[X]}(x)=3|X|-2$. So $|X|$ is even. Lemma 1.1 implies that $G$ has a perfect matching, say $M$. Then $ab\in M$ if and only if $cd\in M$. Since $\partial{(X\setminus\{a\})}=(\partial{(a)\setminus\{ab\}})\cup\{cd\}$, $|M\cap\partial{(X\setminus\{a\})}|=1$. So $\partial{(X\setminus\{a\})}$ is a tight cut of $G$. (Fig. 1(b) illustrates an example).

(ii) Since $G$ is matching covered (Lemma 1.1), $G$ has a perfect matching containing $ab$, say $M$. Then $\{ab,cd\}\subseteq M$ by the discussion in (i). So $M\setminus\{ab,cd\}$ is a perfect matching of $G-\{a,b,c,d\}$, and thus $G-\{a,b,c,d\}$ consists of two vertex-disjoint subgraphs $G[X\setminus\{a,c\}]$ and $G[\overline{X}\setminus\{b,d\}]$ which each has a perfect matching.

(iii) Set $G^{\prime}=G[X]+ac$. If $u\in X\setminus\{a\}$, since $\partial{(X\setminus\{a\})}$ is a tight cut by (i) and $G/(\overline{X}\cup\{a\})$ is simple, $u$ is nice in $G$ by Lemma 2.12. If $u=a$, then $N_{G}[a]=(N_{G^{\prime}}[a]\setminus\{c\})\cup\{b\}$. Since $u$ is nice in $G^{\prime}$, $G^{\prime}-N_{G^{\prime}}[a]$ has a perfect matching, say $M^{\prime}$. On the other hand, (ii) shows that $G[\overline{X}\setminus\{b,d\}]$ has a perfect matching, say $M^{\prime\prime}$. Then $M^{\prime}\cup\{cd\}\cup M^{\prime\prime}$ is a perfect matching of $G-N_{G}[a]$. Therefore, $u$ is nice in $G$.

(iv) If $ac\in E(G)$, then we are done. So suppose that $ac\notin E(G)$. Let $(A_{X},B_{X})$ be a bipartition of $G[X]$ and $l=|E(A_{X},\overline{X})|$. Then $l=0,1,2$. Since $G$ is cubic, $|E(G[X])|=3|A_{X}|-l=3|B_{X}|-(2-l)$. So $l=1$ and $|A_{X}|=|B_{X}|$, which implies that $a$ and $c$ are in distinct color classes of $G[X]$. ∎

Lemma 2.15**.**

Let $G$ be a 2-connected non-bipartite cubic graph with a 2-cut. Then $G$ has a 2-cut $\partial{(X)}=\{ab,cd\}$ such that $a,c\in X$ and $G[X]+ac$ is a 2-connected non-bipartite simple cubic graph with the minimum number of vertices.

Proof.

Assume that $\partial{(Y)}=\{a^{\prime}b^{\prime},c^{\prime}d^{\prime}\}$ is a 2-cut of $G$ such that $a^{\prime},c^{\prime}\in Y$. We claim that at least one of $G[Y]$ and $G[\overline{Y}]$ is non-bipartite. Otherwise, by Lemma 2.14(iv), $a^{\prime}$ and $c^{\prime}$ (resp. $b^{\prime}$ and $d^{\prime}$) lie in different color classes of $G[Y]$ (resp. $G[\overline{Y}]$). Then $G$ is bipartite by Lemma 2.13, a contradiction. Choose a 2-cut $\partial{(Y)}$ so that $G[Y]$ is a non-bipartite subgraph with the minimum number of vertices. Then $a^{\prime}c^{\prime}\notin E(G)$. Otherwise, $\partial{(Y\setminus\{a^{\prime},c^{\prime}\})}$ is also a 2-cut of $G$ as $G$ is cubic. Since $G[Y]$ is non-bipartite and $G[\{a^{\prime},c^{\prime}\}]$ ($=K_{2}$) is bipartite, we deduce that $G[Y\setminus\{a^{\prime},c^{\prime}\}]$ is non-bipartite by Lemmas 2.14(iv) and 2.13, a contradiction to the minimality of $|Y|$. So $G[Y]+a^{\prime}c^{\prime}$ is a 2-connected non-bipartite simple cubic graph. Further, $|V(G[Y]+a^{\prime}c^{\prime})|$ is minimum. Otherwise, there exists a 2-cut $\partial{(Z)}=\{a^{\prime\prime}b^{\prime\prime},c^{\prime\prime}d^{\prime\prime}\}$ such that $a^{\prime\prime},c^{\prime\prime}\in Z$ and $G[Z]+a^{\prime\prime}c^{\prime\prime}$ is a 2-connected non-bipartite simple cubic graph with $|Z|<|Y|$. Lemma 2.14(iv) implies that $G[Z]$ is also a non-bipartite subgraph of $G$, contradicting the choice of $\partial{(Y)}$. Therefore, $\partial{(Y)}$ is a desired 2-cut and the lemma holds. ∎

3 Nice vertices in non-bipartite cubic graphs

Before stating our results, we introduce some necessary notations. Let $G_{1}$ and $G_{2}$ be two vertex-disjoint graphs. Construct two new graphs as follows. Suppose that $u\in V(G_{1})$ and $v\in V(G_{2})$ are two vertices with the same degree. Let $\phi$ be a given bijection between $\partial_{G_{1}}{(u)}$ and $\partial_{G_{2}}{(v)}$. The splicing $G_{1}$ at $u$ and $G_{2}$ at $v$, denoted by $(G_{1}\odot G_{2})_{u,v,\phi}$, is the graph obtained from the union of $G_{1}-u$ and $G_{2}-v$ by joining, for each edge $e\in\partial_{G_{1}}{(u)}$, the end of $e$ in $V(G_{1}-u)$ to the end of $\phi(e)$ in $V(G_{2}-v)$. If $G_{1}$ is a cubic graph and $G_{2}\in\{K_{4},K_{3,3}\}$, then, up to isomorphism, $(G_{1}\odot G_{2})_{u,v,\phi}$ depends only on the choice of $u$ in $G_{1}$, and we denote it simply by $(G_{1}\odot G_{2})_{u}$ (or $G_{1}\odot G_{2}$ if the choice of $u$ is irrelevant). For example, $K_{4}\odot K_{4}=\overline{C_{6}}$, $\overline{C_{6}}\odot K_{4}=R_{8}$ and $K_{3,3}\odot K_{4}=K_{3,3}^{\triangle}$ (see Fig. 2(a)-(c)).

Suppose that $e_{1}=x_{1}x_{2}\in E(G_{1})$ and $e_{2}=y_{1}y_{2}\in E(G_{2})$. The edge splicing $G_{1}$ at $e_{1}$ and $G_{2}$ at $e_{2}$, denoted by $(G_{1}\oplus G_{2})_{e_{1},e_{2}}$ (or simply $(G_{1}\oplus G_{2})_{e_{1}}$ if no confusion occurs), is the graph obtained from the union of $G_{1}-e_{1}$ and $G_{2}-e_{2}$ by identifying $x_{i}$ and $y_{i}$ into a single vertex for $i=1,2$.

We can see that $K_{3,3}^{\triangle}$ has exactly six nice vertices (black) and two non-nice vertices (white); see Fig. 2(c). Let $\widehat{\mathcal{H}}$ be the set of 3-connected cubic bipartite graphs, $\mathcal{G}_{1}=\{(K_{3,3}^{\triangle}\odot H)_{u,v}|u~\text{is not nice in}~K_{3,3}^{\triangle}$ and $H\in\widehat{\mathcal{H}}\}$ and $\mathcal{G}_{2}=\{((K_{3,3}^{\triangle}\odot H_{1})_{u_{1},v_{1}}\odot H_{2})_{u_{2},v_{2}}|u_{i}~\text{is not nice in}~K_{3,3}^{\triangle},H_{i}\in\widehat{\mathcal{H}}$ and $i=1,2\}$. For example, see Fig. 3(a) and (b).

Let $L_{n}$ be a linear quadrangular chain with $n\geq 1$ quadrangles (see Fig. 4(a) for $L_{3}$) and $\mathcal{H}$ be the set of cubic bipartite graphs. We refer to an edge with two end vertices of degree 2 as a 22-edge. Set $\mathcal{H^{\diamond}}=\{(L_{n}\oplus H)_{e,f}|n\geq 1,e$ is a 22-edge and $H\in\mathcal{H}$}. Then every graph in $\mathcal{H^{\diamond}}$ is bipartite and has precisely one 22-edge. To replace an edge $e$ of $K_{4}$ by a graph $H\in\mathcal{H^{\diamond}}$ is the graph obtained by edge splicing $K_{4}$ at $e$ and $H$ at the 22-edge. For $1\leq i\leq 6$, let $\mathcal{F}_{i}$ be the set of graphs obtained from $K_{4}$ by replacing $i$ edges of $K_{4}$ by $i$ graphs in $\mathcal{H^{\diamond}}$ (not necessarily distinct). Fig. 4(b) and (c) illustrate an example in $\mathcal{H^{\diamond}}$ and $\mathcal{F}_{2}$ respectively.

Let $\Upsilon(G)$ denote the number of nice vertices of a cubic graph $G$. It was shown in [21] that non-bipartite cubic graphs on at most 6 vertices are $K_{4}$ and $\overline{C_{6}}$. Clearly, $\Upsilon(K_{4})=4$, $\Upsilon(\overline{C_{6}})=6$ and $\Upsilon(K_{3,3}^{\triangle})=6$. Now we can state our main theorem of this section.

Theorem 3.1**.**

*Let $G$ be a 2-connected non-bipartite cubic graph. Then

(i) $\Upsilon(G)\geq 4$. Moreover, $\Upsilon(G)=4$ if and only if $G\in\{K_{4}\}\cup(\cup_{i=1}^{6}\mathcal{F}_{i})$.

(ii) If $G$ is 3-connected and $G\neq K_{4}$, then $\Upsilon(G)\geq 6$. Moreover, $\Upsilon(G)=6$ if and only if $G\in\{\overline{C_{6}},K_{3,3}^{\triangle}\}\cup\mathcal{G}_{1}\cup\mathcal{G}_{2}$.*

In order to prove Theorem 3.1, we first prove several auxiliary results. A nontrivial barrier $S$ of a graph with a perfect matching is minimal if no proper subset of $S$ is also a nontrivial barrier.

Lemma 3.2**.**

Let $G$ be a 3-connected non-bicritical non-bipartite cubic graph. Then $G$ has a minimal nontrivial barrier $S$ so that each vertex of $S$ is nice.

Proof.

We proceed by induction on $|V(G)|$. By Lemma 1.1, $G$ is matching covered. Since $G$ is non-bicritical, $G$ has a nontrivial barrier by Lemma 2.2(i). Take a minimal nontrivial barrier $S$. Since $G$ is 3-connected, $|S|\geq 3$. By Lemma 2.11(iii), $G-S$ has a nontrivial non-bipartite component $K$, which is odd by Lemma 2.2(ii). So $|V(G)|\geq 8$. If $|V(G)|=8$, then $|S|=3$ and $K$ is a triangle. So $G\cong K_{3,3}^{\triangle}$ and each vertex of $S$ is nice. Now assume that $|V(G)|\geq 10$. Let $G^{\prime}$ be the graph obtained from $G$ by shrinking each nontrivial component of $G-S$ other than $K$ into a single vertex.

Claim 1. $G_{2}^{\prime}:=G^{\prime}/(V(K)\rightarrow k)$ is a brace.

Proof.

By Lemma 2.2(iii), $S$ is independent. So $G_{2}^{\prime}$ is bipartite. By Lemma 2.11(ii), $G_{2}^{\prime}$ is a 3-connected simple cubic graph and so is matching covered. If $G_{2}^{\prime}$ is not a brace, then $G_{2}^{\prime}$ has a nontrivial tight cut, say $\partial_{G_{2}^{\prime}}{(Y)}$, by Lemma 2.8. Since $|Y|$ is odd by Lemma 2.7, either $Y_{+}\subseteq S$ or $\overline{Y}_{+}\subseteq S$, where $\overline{Y}=V(G_{2}^{\prime})\setminus Y$. Adjust notation so that $Y_{+}\subseteq S$. Then $|Y_{+}|=|Y\cap S|=|Y_{-}|+1=|Y\setminus S|+1$ by Lemma 2.7 again. Since $\partial_{G_{2}^{\prime}}{(Y)}$ is a 3-cut by Lemma 2.6 and $G_{2}^{\prime}$ is also 3-edge-connected, $G_{2}^{\prime}[\overline{Y}]$ is connected. Then $o(G_{2}^{\prime}-(S\cap Y))=|Y\setminus S|+1=|S\cap Y|$, which implies $o(G-(S\cap Y))=|S\cap Y|$. So $S\cap Y$ is a nontrivial barrier of $G$, a contradiction to the minimality of $S$. Hence $G_{2}^{\prime}$ is a brace. ∎

By Lemma 2.11(ii), $G_{1}:=G/(\overline{V(K)}\rightarrow\overline{k})$ is a 3-connected non-bipartite simple cubic graph. Next we consider two cases according to whether $\overline{k}$ is nice in $G_{1}$ or not.

Case 1. $\overline{k}$ is nice in $G_{1}$.

We will show that $S$ is a desired barrier. By using Lemmas 2.11(i), (ii) and 2.12 repeatedly, we only need to show that each vertex of $S$ is nice in $G^{\prime}$. By Lemma 2.11(i), $\partial{(V(K))}$ is a matching of size 3. Set $\partial{(V(K))}=\{u_{i}v_{i}|u_{i}\in S,v_{i}\in V(K)~\text{and}~i=1,2,3\}$. Let $w$ be a vertex in $S$ and $N_{G^{\prime}}(w)=\{w_{1},w_{2},w_{3}\}$. By Claim 1 and Lemma 2.9, $G_{2}^{\prime}-\{u_{2},u_{3},w_{2},w_{3}\}$ has a perfect matching $M_{2}$. Since $\overline{k}$ is nice in $G_{1}$, $G_{1}-N_{G_{1}}[\overline{k}]$ has a perfect matching $M_{1}$ (see Fig. 5(a)). If $w\in\{u_{1},u_{2},u_{3}\}$, assume that $w=u_{1}$ and $w_{1}=v_{1}$, then $wk\in M_{2}$ (see Fig. 5(b)) and $G^{\prime}-N_{G^{\prime}}[w]=(G_{1}-\{\overline{k},v_{1}\})\cup(G_{2}^{\prime}-\{w,k,w_{2},w_{3}\})\cup\{u_{2}v_{2},u_{3}v_{3}\}$. So $M_{1}\cup(M_{2}\setminus\{wk\})\cup\{u_{2}v_{2},u_{3}v_{3}\}$ is a perfect matching of $G^{\prime}-N_{G^{\prime}}[w]$ (see Fig. 5(c)). If $w\notin\{u_{1},u_{2},u_{3}\}$, then $\{u_{1}k,ww_{1}\}\subseteq M_{2}$ (see Fig. 5(d)) and $G^{\prime}-N_{G^{\prime}}[w]=(G_{1}-\overline{k})\cup(G_{2}^{\prime}-(\{k\}\cup N_{G^{\prime}}[w]))\cup\partial{(V(K))}$. So $M_{1}\cup(M_{2}\setminus\{u_{1}k,ww_{1}\})\cup\partial{(V(K))}$ is a perfect matching of $G^{\prime}-N_{G^{\prime}}[w]$. Hence $w$ is nice in $G^{\prime}$. By the arbitrariness of $w$, each vertex of $S$ is nice in $G^{\prime}$.

Case 2. $\overline{k}$ is not nice in $G_{1}$.

Lemma 1.2 implies that $G_{1}$ is not bicritical. So $G_{1}$ has a minimal nontrivial barrier $S_{1}$ so that each vertex of $S_{1}$ is nice in $G_{1}$ by the induction hypothesis. Then $\overline{k}\notin S_{1}$ and $S_{1}\subseteq V(K)$. Since $\partial{(V(K))}$ is a tight cut of $G$ by Lemma 2.11(i), each vertex of $S_{1}$ is also nice in $G$ by Lemma 2.12. Since $G_{2}:=G/((V(K)\rightarrow k)$ is 3-connected (Lemma 2.11(ii)), $G_{2}-k$ is connected. Then $G_{2}-k$ is contained in some component of $G-S_{1}$, which implies $o(G-S_{1})=o(G_{1}-S_{1})$. So $S_{1}$ is also a nontrivial barrier of $G$. If $S_{1}$ properly contains a nontrivial barrier $T$ of $G$, then $o(G-T)=o(G_{1}-T)$ by the same reason as above. Hence $T$ is also a barrier of $G_{1}$, contradicting the minimality of $S_{1}$. So $S_{1}$ is a minimal nontrivial barrier of $G$, which is a desired barrier. ∎

Lemma 3.3**.**

Let $G$ be a 2-connected non-bipartite cubic graph. Then $\Upsilon(G)\geq 4$. Moreover, $\Upsilon(G)\geq 6$ if $G$ is 3-connected and $G\neq K_{4}$.

Proof.

We use induction on $|V(G)|$. If $|V(G)|=4$, then $G=K_{4}$ and $\Upsilon(G)=4$. Suppose that $|V(G)|\geq 6$ and the result holds for graphs with fewer vertices than $|V(G)|$. If $G$ is bicritical, then $\Upsilon(G)=|V(G)|\geq 6$ by Lemma 1.2 and the result holds. So suppose that $G$ is not bicritical. First assume that $G$ is 3-connected. By Lemma 3.2, $G$ has a minimal nontrivial barrier $S$ so that each vertex of $S$ is nice. Since $G$ is 3-connected, $|S|\geq 3$. Further, since $G$ is non-bipartite, $G-S$ contains a nontrivial non-bipartite component $K$ by Lemma 2.11(iii). Since $\partial{(V(K))}$ is a tight cut by Lemma 2.11(i), $\Upsilon(G)\geq\Upsilon(G/\overline{V(K)})-1+|S|\geq\Upsilon(G/\overline{V(K)})+2$ by using Lemma 2.12. By Lemma 2.11(ii), $G/\overline{V(K)}$ is a 3-connected non-bipartite simple cubic graph. So $\Upsilon(G/\overline{V(K)})\geq 4$ by the induction hypothesis and thus $\Upsilon(G)\geq 6$.

Now suppose that $G$ is not 3-connected. Since a cubic graph is 3-connected if and only if it is 3-edge-connected, $G$ has a 2-cut. By Lemma 2.15, we can take a 2-cut $\partial{(X)}=\{ab,cd\}$ such that $a,c\in X$ and $G[X]+ac$ is a 2-connected non-bipartite simple cubic graph. Then $\Upsilon(G[X]+ac)\geq 4$ by the induction hypothesis. Further, as $ac\notin E(G)$, $\Upsilon(G)\geq\Upsilon(G[X]+ac)\geq 4$ by using Lemma 2.14(iii), and so the proof is completed. ∎

In the following, we show that the lower bounds in Lemma 3.3 are sharp.

Lemma 3.4**.**

Let $G$ be a 3-connected non-bipartite cubic graph. Then $\Upsilon(G)=6$ if and only if $G\in\{\overline{C_{6}},K_{3,3}^{\triangle}\}\cup\mathcal{G}_{1}\cup\mathcal{G}_{2}$.

Proof.

Sufficiency. If $G\in\{\overline{C_{6}},K_{3,3}^{\triangle}\}$, then $\Upsilon(G)=6$ clearly. Assume that $G=(K_{3,3}^{\triangle}\odot H)_{u,v}\in\mathcal{G}_{1}$, where $H\in\mathcal{\widehat{H}}$. Let $H=H(A_{H},B_{H})$ and $v\in B_{H}$. Then $A_{H}$ (resp. $B_{H}\cup N_{K_{3,3}^{\triangle}}(u)$) is a barrier of $G$ such that each vertex of $G$ in $B_{H}\setminus\{v\}$ (resp. $A_{H}\cup\{u^{\prime}\}$) is an isolated vertex, where $u^{\prime}$ ($\neq u$) is the other non-nice vertex of $K_{3,3}^{\triangle}$. By Theorem 2.3, each vertex of $G$ in $(V(H)\setminus\{v\})\cup\{u^{\prime}\}$ is not nice. Since $\partial{(\overline{V(H-v)})}$ is a tight cut of $G$ and $G/V(H-v)\cong K_{3,3}^{\triangle}$, the 6 nice vertices of $K_{3,3}^{\triangle}$ are also nice in $G$ by Lemma 2.12. So $\Upsilon(G)=6$. Similarly, we can prove that $\Upsilon(G)=6$ if $G\in\mathcal{G}_{2}$. Hence the sufficiency holds.

Necessity. If $G$ is bicritical, then $|V(G)|=6$ by Lemma 1.2 and so $G=\overline{C_{6}}$. Next suppose that $G$ is not bicritical. By Lemma 3.2, $G$ has a minimal nontrivial barrier $S$ so that each its vertex is nice. Lemma 2.11(iii) implies that $G-S$ contains a nontrivial non-bipartite component $K$. Then $G/\overline{V(K)}$ is a 3-connected non-bipartite simple cubic graph by Lemma 2.11(ii). Combining Lemmas 2.11(i) and 2.12, $\Upsilon(G)\geq\Upsilon(G/\overline{V(K)})-1+|S|\geq\Upsilon(G/\overline{V(K)})+2$, where the second inequality is because $G$ is 3-connected. As $\Upsilon(G)=6$, $\Upsilon(G/\overline{V(K)})\leq 4$. By Lemma 3.3, $G/\overline{V(K)}=K_{4}$, which implies that $K$ is a triangle. Since all vertices of $K$ are nice in $G/\overline{V(K)}$, $G$ has exactly 3 nice vertices in $K$ by Lemma 2.12. So $|S|=3$ and $G$ has no nice vertices in $G-S-V(K)$. By Lemmas 1.1 and 2.2(ii), $G-S$ has no even components. As $o(G-S)=|S|=3$, $G-S$ has precisely 3 components. We claim that the other nontrivial components of $G-S$ is bipartite. Otherwise, say $K^{\prime}$ is another nontrivial non-bipartite component of $G-S$. Then we have that $K^{\prime}$ contains at least 3 nice vertices of $G$ by Lemmas 2.11(i), (ii), 3.3 and 2.12, a contradiction. If $K$ is the unique nontrivial component of $G-S$, then $G\cong K_{3,3}^{\triangle}$. Assume that $G-S$ has exactly two nontrivial components $K$ and $K^{*}$. Set $G_{1}=G/(V(K^{*})\rightarrow k^{*})$ and $G_{2}=G/(\overline{V(K^{*})}\rightarrow\overline{k^{*}})$. Then we have that $G_{1}\cong K_{3,3}^{\triangle}$, $G_{2}\in\mathcal{\widehat{H}}$ by Lemma 2.11(ii) and $k^{*}$ is not nice in $G_{1}$. So $G=(G_{1}\odot G_{2})_{k^{*},\overline{k^{*}}}\in\mathcal{G}_{1}$. Similarly, $G\in\mathcal{G}_{2}$ if $G-S$ consists of three nontrivial components. Hence $G\in\{\overline{C_{6}},K_{3,3}^{\triangle}\}\cup\mathcal{G}_{1}\cup\mathcal{G}_{2}$. ∎

Lemma 3.5**.**

Let $G$ be a 2-connected non-bipartite cubic graph. Then $\Upsilon(G)=4$ if and only if $G\in\{K_{4}\}\cup(\cup_{i=1}^{6}\mathcal{F}_{i})$.

Proof.

If $|V(G)|=4$, then $G=K_{4}$ and the result holds. So assume that $|V(G)|\geq 6$.

Sufficiency. Assume that $G$ is a graph in $\mathcal{F}_{i}$ for some $1\leq i\leq 6$, which is obtained by replacing $i$ edges $e_{1},\ldots,e_{i}$ of $K_{4}$ by $i$ graphs $H_{1}^{\diamond},\ldots,H_{i}^{\diamond}$ in $\mathcal{H^{\diamond}}$ respectively. For $1\leq j\leq i$, let $(X_{j1},X_{j2})$ be the bipartition of $H_{j}^{\diamond}$ and $x_{j1}x_{j2}$ be its 22-edge with $x_{jl}\in X_{jl}$ for $l=1,2$, and $e_{j}=y_{j1}y_{j2}$. Denote by $z_{jl}$ the vertex obtained by identifying $x_{jl}$ and $y_{jl}$ for $l=1,2$. Then $(X_{j1}\setminus\{x_{j1}\})\cup\{z_{j1}\}$ is a barrier of $G$ such that vertices of $G$ in $X_{j2}\setminus\{x_{j2}\}$ are isolated vertices, which are not nice in $G$ by Theorem 2.3. By symmetry, vertices of $G$ in $X_{j1}\setminus\{x_{j1}\}$ are not nice. Moreover, by using Lemma 2.14(iii) $i$ times, we obtain that the 4 vertices of $G$ in $\overline{\cup_{j=1}^{i}V(H_{j}^{\diamond})\setminus\{x_{j1},x_{j2}\}}$ are nice, and so $\Upsilon(G)=4$.

Necessity. We first prove the following claim.

Claim 2. $G$ has a 2-cut $\partial{(X)}=\{ab,cd\}$ such that $a,c\in X$ and

(i) $G_{1}:=G[X]+ac$ is a 2-connected non-bipartite simple cubic graph with the minimum number of vertices,

(ii) $\Upsilon(G_{1})=4$, and either $G_{1}\cong K_{4}$ or $G_{1}$ has a 2-cut,

(iii) $G_{2}:=G[\overline{X}\cup\{a,c\}]+ac\in\mathcal{H^{\diamond}}$.

Proof.

From Lemma 3.3, $G$ has a 2-cut. By Lemma 2.15, we can take a 2-cut $\partial{(X)}=\{ab,cd\}$ of $G$ such that $a,c\in X$ and Claim 2(i) holds. Then $ac\notin E(G)$. Using Lemma 2.14(iii), $\Upsilon(G)\geq\Upsilon(G_{1})$. As $\Upsilon(G)=4$, Lemma 3.3 implies that $\Upsilon(G_{1})=4$, and either $G_{1}\cong K_{4}$ or $G_{1}$ has a 2-cut. So Claim 2(ii) holds. It remains to prove Claim 2(iii). Since $G$ is cubic, $G_{2}=(L_{n}\oplus H)_{e,f}$ for some $n\geq 1$ and a 2-connected cubic graph $H$, where $e$ is a 22-edge of $L_{n}$. If $H$ is non-bipartite, then $\Upsilon(H)\geq 4$ by Lemma 3.3, and so $\Upsilon(G)\geq\Upsilon(G_{1})+\Upsilon(G_{2})\geq 8$ by Lemma 2.14(iii), a contradiction. Thus $H$ is bipartite and $G_{2}\in\mathcal{H}^{\diamond}$. ∎

By Claim 2, take a 2-cut $\partial{(X)}=\{ab,cd\}$ of $G$ such that $a,c\in X$ and Claim 2(i)-(iii) holds. We proceed by induction on the number of 2-cuts of $G$. If $\partial{(X)}$ is the unique 2-cut of $G$, then either $G_{1}\cong K_{4}$ or $G_{1}$ has a 2-cut by Claim 2(ii). If $G_{1}$ has a 2-cut, say $\partial_{G_{1}}{(Y)}$, then $G$ has another 2-cut $C$, where $C=\partial{(Y)}$ if $ac\notin E(G_{1}[Y])$, and $C=\partial{(\overline{Y})}$ otherwise, a contradiction. So $G_{1}\cong K_{4}$ and $G$ is obtained by replacing exactly one edge in $G_{1}$ by the graph $G_{2}$ in $\mathcal{H}^{\diamond}$ by Claim 2(iii). Thus $G\in\mathcal{F}_{1}$ and the result holds. Assume that the result holds for graphs with less 2-cuts than $G$. By Claim 2(ii) again, if $G_{1}\cong K_{4}$, then $G\in\mathcal{F}_{1}$ due to Claim 2(iii), and otherwise $G_{1}\in\cup_{i=1}^{6}\mathcal{F}_{i}$ by induction. Assume that $G_{1}\in\mathcal{F}_{i}$ for some $1\leq i\leq 6$, which is obtained by replacing $i$ edges $f_{1},f_{2},\ldots,f_{i}$ of $K_{4}$ by $i$ graphs $H_{1},H_{2},\ldots,H_{i}$ in $\mathcal{H^{\diamond}}$, respectively. Let $U_{j}$ be the set of vertices in $H_{j}$ with degree 3 and $V_{j}=V(G_{1})\setminus U_{j}$ for $1\leq j\leq i$. We claim that $ac\in E(K_{4})$. Otherwise, $ac\in\partial_{G_{1}}{(V_{j})}\cup E(G_{1}[U_{j}])$ for some $1\leq j\leq i$. Since $G_{1}[U_{j}]$ is bipartite and $G_{1}$ is non-bipartite, $G_{1}[V_{j}]$ ($=G[V_{j}]$) is non-bipartite by Lemma 2.13. Then $\partial{(V_{j})}$ is a 2-cut of $G$ and $G[V_{j}]+f_{j}^{\prime}$ is a 2-connected non-bipartite simple cubic graph with less vertices than $G_{1}$, a contradiction to Claim 2(i), where $f_{j}^{\prime}$ is the edge corresponding to the edge $f_{j}$ in $K_{4}$. So $G_{1}\in\mathcal{F}_{i}$ for some $1\leq i\leq 5$ and $G\in\mathcal{F}_{i+1}\subseteq\cup_{i=2}^{6}\mathcal{F}_{i}$, which completes the proof. ∎

Consequently, Theorem 3.1 is obtained by Lemmas 3.3-3.5.

4 Nice pairs in bipartite cubic graphs

In this section, we discuss nice pairs of vertices in cubic bipartite graphs. Let $\mathcal{T}$ be the set of cubic bipartite graphs defined as follows: (i) $K_{3,3}\in\mathcal{T}$, (ii) if $T\in\mathcal{T}$, then $(T\oplus(L_{n}\oplus K_{3,3})_{e_{1}})_{f,e_{2}}\in\mathcal{T}$, where $e_{1}$ and $e_{2}$ are two 22-edges of $L_{n}$ and $n\geq 1$. For a cubic bipartite graph $G(A,B)$, denote by $(a,b)$ (resp. $(A^{\prime},B^{\prime}))$ a pair of vertices $a\in A$ and $b\in B$ (resp. vertex subsets $A^{\prime}\subseteq A$ and $B^{\prime}\subseteq B$). Then our main results are as follows.

Theorem 4.1**.**

Let $G(A,B)$ be a cubic bipartite graph. Then $G$ has a nice pair set $(A^{\prime},B^{\prime})$ with $|A^{\prime}|\geq 3$ and $|B^{\prime}|\geq 3$. Moreover, for each nice pair set $(A^{\prime},B^{\prime})$, $|A^{\prime}|\leq 3$ and $|B^{\prime}|\leq 3$ if and only if $G\in\mathcal{T}$.

Corollary 4.2**.**

Let $G$ be a cubic bipartite graph. Then $G$ has at least 9 nice pairs of vertices, with equality if and only if $G=K_{3,3}$.

Before proving Theorem 4.1 and Corollary 4.2, we show some useful results. Since the edge chromatic number of a bipartite graph is equal to its maximum degree (see [1], Theorem 17.2), every cubic bipartite graph has 3 edge-disjoint perfect matchings. This implies the following result.

Lemma 4.3**.**

Every cubic bipartite graph is matching covered.

Proposition 4.4**.**

Let $G(A,B)$ be a cubic bipartite graph. Then $G$ is a brace if and only if each pair $(a,b)$ of $G$ is nice (i.e. $(A,B)$ is a nice pair set).

Proof.

Necessity. Set $N_{G}(a)=\{a_{1},a_{2},a_{3}\}$ and $N_{G}(b)=\{b_{1},b_{2},b_{3}\}$. By Lemma 2.9, $G-\{a_{2},a_{3},b_{2},b_{3}\}$ has a perfect matching, denoted by $M$. If $a\notin N_{G}(b)$, then $\{aa_{1},bb_{1}\}\subseteq M$ since $G$ is cubic. It follows that $M\setminus\{aa_{1},bb_{1}\}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$. If $a\in N_{G}(b)$, assume that $a=b_{1}$ and $b=a_{1}$, then $ab\in M$ and so $M\setminus\{ab\}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$. Thus $(a,b)$ is a nice pair and the necessity holds.

Sufficiency. Suppose to the contrary that $G$ is not a brace. Since $G$ is matching covered (Lemma 4.3) and bipartite, $G$ has a nontrivial tight cut by Lemma 2.8. Take a nontrivial tight cut $\partial{(X)}$ so that $X_{+}\subseteq A$. By Lemma 2.7, $|A\cap X|=|B\cap X|+1$ and $E(B\cap X,A\cap\overline{X})=\emptyset$. For $a\in A\cap\overline{X}$ and $b\in B\cap X$, since $(a,b)$ is nice in $G$, $G-N_{G}[a]-N_{G}[b]$ has a perfect matching, say $F$. As $|(A\cap X)\setminus N_{G}(b)|=|A\cap X|-3=|B\cap X|-2<|(B\cap X)\setminus\{b\}|$, $F\cap E(B\cap X,A\cap\overline{X})\neq\emptyset$, a contradiction. So $G$ is a brace. ∎

Lemma 4.5**.**

*Let $G(A,B)$ be a cubic bipartite graph. Assume that $\partial{(X)}$ is a nontrivial tight cut of $G$ such that $|A\cap X|=|B\cap X|+1$ and $G/\overline{X}$ is simple. For $a\in A$ and $b\in B$,

(i) if $a,b\in X$, then $(a,b)$ is a nice pair of $G/\overline{X}$ if and only if it is a nice pair of $G$,

(ii) Assume that $G$ is 3-connected, $a\in X$ and $b\in\overline{X}$. If $(a,\overline{x})$ and $(x,b)$ are nice pairs of $G/(\overline{X}\rightarrow\overline{x})$ and $G/(X\rightarrow x)$ respectively, then $(a,b)$ is a nice pair of $G$.*

Proof.

Set $G_{1}=G/(\overline{X}\rightarrow\overline{x})$ and $G_{2}=G/(X\rightarrow x)$. For an edge $e\in\partial{(X)}$, denote by $e_{i}$ the edge in $G_{i}$ corresponding to $e$ for $i=1,2$.

(i) Necessity. Since $(a,b)$ is a nice pair of $G_{1}$, $G_{1}-N_{G_{1}}[a]-N_{G_{1}}[b]$ has a perfect matching, say $M_{1}$. As $G_{1}$ is simple and $b\neq\overline{x}$, $(M_{1}\cup\partial_{G_{1}}{(a)})\cap\partial_{G_{1}}{(\overline{x})}$ consists of exactly one edge, say $e_{1}$. Since $G$ is matching covered by Lemma 4.3, $G_{2}$ is also matching covered. So the graph obtained from $G_{2}$ by removing the two end-vertices of $e_{2}$ has a perfect matching, denoted by $M_{2}$. If $e_{1}\in M_{1}$, then $G-N_{G}[a]-N_{G}[b]=(G_{1}-N_{G_{1}}[a]-N_{G_{1}}[b]-\{\overline{x}\})\cup(G_{2}-x)$, and so $(M_{1}\setminus\{e_{1}\})\cup M_{2}\cup\{e\}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$. If $e_{1}\notin M_{1}$, then $M_{1}\cup M_{2}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$. Thus $(a,b)$ is a nice pair of $G$.

Sufficiency. Let $M$ be a perfect matching of $G-N_{G}[a]-N_{G}[b]$. Then $(M\cup\partial{(a)})\cap\partial{(X)}$ consists of precisely one edge, say $e^{\prime}$. If $e^{\prime}\in M$, then $(M\cap E(G_{1}))\cup\{e_{1}^{\prime}\}$ is a perfect matching of $G_{1}-N_{G_{1}}[a]-N_{G_{1}}[b]$. If $e^{\prime}\notin M$, then $M\cap E(G_{1})$ is a perfect matching of $G_{1}-N_{G_{1}}[a]-N_{G_{1}}[b]$. So $(a,b)$ is a nice pair of $G_{1}$.

(ii) By Lemmas 2.5 and 2.6, $\partial{(X)}$ is a matching of size 3. Assume that $\partial{(X)}=\{\overline{x}_{i}x_{i}|\overline{x}_{i}\in A,x_{i}\in B~\text{and}~i=1,2,3\}$. Then $\overline{x}_{i}\overline{x}\in E(G_{1})$ and $xx_{i}\in E(G_{2})$ for $i=1,2,3$. Let $F_{1}$ and $F_{2}$ be perfect matchings of $G_{1}-N_{G_{1}}[a]-N_{G_{1}}[\overline{x}]$ and $G_{2}-N_{G_{2}}[x]-N_{G_{2}}[b]$, respectively. If $a\notin N_{G_{1}}(\overline{x})$ and $b\notin N_{G_{2}}(x)$, then obviously $F_{1}\cup F_{2}\cup\partial{(X)}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$. If $a\notin N_{G_{1}}(\overline{x})$ and $b\in N_{G_{2}}(x)$, assume that $b=x_{3}$, then $G-N_{G}[a]-N_{G}[b]=(G_{1}-N_{G_{1}}[a]-\{\overline{x}_{3},\overline{x}\})\cup(G_{2}-N_{G_{2}}[b])$, and so $F_{1}\cup F_{2}\cup\{\overline{x}_{1}x_{1},\overline{x}_{2}x_{2}\}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$ (see Fig. 6(a)). If $a\in N_{G_{1}}(\overline{x})$ and $b\notin N_{G_{2}}(x)$, then $G-N_{G}[a]-N_{G}[b]$ also has a perfect matching by symmetry. Now assume that $a\in N_{G_{1}}(\overline{x})$ and $b\in N_{G_{2}}(x)$. If $ab\notin E(G)$, say $a=\overline{x}_{1}$ and $b=x_{3}$, then $G-N_{G}[a]-N_{G}[b]=(G_{1}-N_{G_{1}}[a]-\overline{x}_{3})\cup(G_{2}-N_{G_{2}}[b]-x_{1})$, and so $F_{1}\cup F_{2}\cup\{\overline{x}_{2}x_{2}\}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$ (see Fig. 6(b)). Similarly, if $ab\in E(G)$, say $a=\overline{x}_{1}$ and $b=x_{1}$, then $F_{1}\cup F_{2}\cup\{\overline{x}_{2}x_{2},\overline{x}_{3}x_{3}\}$ is a perfect matching of $G-N_{G}[a]-N_{G}[b]$ (see Fig. 6(c)). Therefore, $(a,b)$ is a nice pair of $G$. ∎

Note that $K_{3,3}$ and $H_{4,4}$ (see Fig. 2(d)) are all 3-connected cubic bipartite graphs on at most 8 vertices (also see [21]).

Lemma 4.6**.**

Let $G(A,B)$ be a 3-connected cubic bipartite graph. If $|V(G)|\geq 10$, then $G$ has a nice pair set $(A^{\prime},B^{\prime})$ with $|A^{\prime}|\geq 3$ and $|B^{\prime}|\geq 5$, and otherwise $(A,B)$ is a nice pair set of $G$.

Proof.

If $|V(G)|<10$, then $G\in\{K_{3,3},H_{4,4}\}$ and each pair of vertices is nice. So $(A,B)$ is a nice pair set and the latter statement holds. Next assume that $|V(G)|\geq 10$. We proceed by induction on the number of nontrivial tight cuts of $G$. If $G$ has no nontrivial tight cuts, since $G$ is matching covered (Lemma 4.3) and bipartite, $G$ is a brace by Lemma 2.8. Since $|A|=|B|\geq 5$, the result follows from Proposition 4.4. Assume that $G$ has a nontrivial tight cut and the result holds for graphs with fewer nontrivial tight cuts than $G$. Lemma 2.8 implies that $G$ is not a brace. By Lemma 2.10, $G$ has a nontrivial tight cut $\partial{(X)}$ so that one of the two $\partial{(X)}$-contractions is a brace. Adjust notation so that $G_{1}:=G/(\overline{X}\rightarrow\overline{x})$ is a brace. By Lemmas 2.4-2.6, $G_{2}:=G/(X\rightarrow x)$ is a 3-connected simple cubic bipartite graph. Set $G_{i}=G_{i}(A_{i},B_{i})$, $i=1,2$. By Proposition 4.4, $(A_{1},B_{1})$ is a nice pair set of $G_{1}$ with $|A_{1}|=|B_{1}|\geq 3$. If $G_{2}$ is a brace, then $(A_{2},B_{2})$ is also a nice pair set of $G_{2}$ with $|A_{2}|=|B_{2}|\geq 3$. Using Lemma 4.5, $(A_{i},B)$ is a nice pair set of $G$ with $|A_{i}|\geq 3$ and $|B|\geq 5$ for some $i=1,2$. If $G_{2}$ is not a brace, then $|V(G_{2})|\geq 10$. By the induction hypothesis, $G_{2}$ has a nice pair set $(A_{2}^{\prime},B_{2}^{\prime})$ with $|A_{2}^{\prime}|\geq 3$ and $|B_{2}^{\prime}|\geq 5$. If $x\in A_{2}^{\prime}$, then $\overline{x}\in B_{1}$ and $(A_{1},B_{2}^{\prime})$ is a nice pair set of $G$ by Lemma 4.5(ii). If $x\in B_{2}^{\prime}$, then $\overline{x}\in A_{1}$ and $(A_{2}^{\prime},B_{1}\cup(B_{2}^{\prime}\setminus\{x\}))$ is a nice pair set of $G$ with $|A_{2}^{\prime}|\geq 3$ and $|B_{1}\cup(B_{2}^{\prime}\setminus\{x\})|\geq 3+4=7$ by Lemma 4.5. If $x\notin A_{2}^{\prime}$ and $x\notin B_{2}^{\prime}$, then $(A_{2}^{\prime},B_{2}^{\prime})$ is a nice pair set of $G$ by Lemma 4.5(i), and so the lemma holds. ∎

Lemma 4.7**.**

*Assume that $\partial{(X)}=\{uv,wz\}$ is a 2-cut of a cubic bipartite graph $G(A,B)$ such that $u,w\in X$ and $uw\notin E(G)$. Then the following two statements hold.

(i) If $(a,b)$ is a nice pair of $G$, then either $\{a,b\}\subseteq X$ or $\{a,b\}\subseteq\overline{X}$,

(ii) If $\{a,b\}\subseteq X$, then $(a,b)$ is a nice pair of $G$ if and only if it is nice in $G[X]+uw$.*

Proof.

Since $G[X]$ ($\subseteq G$) is bipartite, $u$ and $w$ are in different color classes of $G$ by Lemma 2.14(iv). Without loss of generality, assume that $u\in A$. Then $z\in A$ and $v,w\in B$.

(i) Otherwise, suppose that $a\in X$ and $b\in\overline{X}$. Since $(a,b)$ is nice in $G$, $G-N_{G}[a]-N_{G}[b]$ has a perfect matching, denoted by $M$. Let $A_{1}=A\cap X$ and $B_{1}=B\cap X$. Since $G$ is cubic, $|E(G[X])|=3|A_{1}|-1=3|B_{1}|-1$, and so $|A_{1}|=|B_{1}|$. If $a\neq u$, then $|M\cap E(A_{1},\overline{X})|=2$ since $|B_{1}\setminus N_{G}(a)|=|B_{1}|-3=|A_{1}\setminus\{a\}|-2$, which is impossible as $E(A_{1},\overline{X})=\{uv\}$. If $a=u$, then $uv\notin M$ and $|M\cap E(A_{1},\overline{X})|=1$, which is also impossible. So (i) holds.

(ii) Set $G_{1}=G[X]+uw$. Then $G_{1}$ is isomorphic to $G/(\overline{X}\cup\{u\})$ and $G/(\overline{X}\cup\{w\})$. By Lemma 2.14(i), $\partial{(X\setminus\{u\})}$ and $\partial{(X\setminus\{w\})}$ are tight cuts of $G$. If $a\neq u$ or $b\neq w$, then $(a,b)$ is a nice pair of $G$ if and only if it is nice in $G_{1}$ by Lemma 4.5(i). So we only need to show that the result holds for $a=u$ and $b=w$. If $(u,w)$ is a nice pair of $G$, then $G-N_{G}[u]-N_{G}[w]$ has a perfect matching, say $F$. It follows that $F\cap E(G_{1})$ is a perfect matching of $G_{1}-N_{G_{1}}[u]-N_{G_{1}}[w]$. So $(u,w)$ is nice in $G_{1}$. Conversely, if $(u,w)$ is nice in $G_{1}$, then $G_{1}-N_{G_{1}}[u]-N_{G_{1}}[w]$ has a perfect matching, say $F_{1}$. Note that $G-N_{G}[u]-N_{G}[w]=(G_{1}-N_{G_{1}}[u]-N_{G_{1}}[w])\cup(G[\overline{X}\setminus\{v,z\}])$. By Lemma 2.14(ii), $G[\overline{X}\setminus\{v,z\}]$ has a perfect matching, denoted by $F_{2}$. Then $F_{1}\cup F_{2}$ is a perfect matching of $G-N_{G}[u]-N_{G}[w]$. So $(u,w)$ is nice in $G$, and the proof is completed. ∎

Lemma 4.8**.**

*Let $G(A,B)$ be a cubic bipartite graph with a 2-cut. Then

(i) $G$ has a 2-cut $\partial{(X)}=\{uv,wz\}$ such that $u,w\in X$, $G[X]+uw\in\mathcal{\widehat{H}}$ and $G[\overline{X}\cup\{u,w\}]+uw\in\mathcal{H}^{\diamond}$,

(ii) $G$ has a nice pair set $(A^{\prime},B^{\prime})$ with $|A^{\prime}|\geq 3$ and $|B^{\prime}|\geq 3$.*

Proof.

Take a 2-cut $\partial{(X)}=\{uv,wz\}$ of $G$ such that $|X|$ is minimum. Then $uw\notin E(G)$. Otherwise, $\partial{(X\setminus\{u,w\})}$ is also a 2-cut of $G$ as $G$ is cubic with $|X\setminus\{u,w\}|<|X|$, a contradiction. Set $G_{1}=G[X]+uw$ and $G_{2}=G[\overline{X}\cup\{u,w\}]+uw$. Then $G_{1}$ is a simple cubic graph and $G_{2}=(L_{n}\oplus H)_{e,f}$ for some $n\geq 1$, where $e$ is a 22-edge and $H$ is a cubic graph. Lemma 2.14(iv) implies that $G_{1}$ and $H$ are bipartite. So $G_{2}\in\mathcal{H}^{\diamond}$. If $G_{1}$ is not 3-connected, then $G_{1}$ has a 2-cut $\partial_{G_{1}}{(Y)}=\{u_{1}v_{1},w_{1}z_{1}\}$ such that $u_{1},w_{1}\in Y$ and $u_{1}w_{1}\notin E(G_{1})$. But thus $\partial{(Y)}$ is a 2-cut of $G$ if $ac\notin E(G_{1}[Y])$; $\partial{(\overline{Y})}$ is a 2-cut of $G$ otherwise, a contradiction to the minimality of $|X|$. So $G_{1}$ is 3-connected. Hence $G_{1}\in\mathcal{\widehat{H}}$, and (i) holds. By Lemma 4.6, we obtain that $G_{1}$ has a nice pair set $(A^{\prime},B^{\prime})$ with $|A^{\prime}|\geq 3$ and $|B^{\prime}|\geq 3$, which is also nice in $G$ by Lemma 4.7(ii). So (ii) holds. ∎

Next we show that the lower bounds on $|A^{\prime}|$ and $|B^{\prime}|$ of Theorem 4.1 are best possible.

Lemma 4.9**.**

Let $G(A,B)$ be a cubic bipartite graph and $(A^{\prime},B^{\prime})$ be a nice pair set of $G$. Then $|A^{\prime}|\leq 3$ and $|B^{\prime}|\leq 3$ if and only if $G\in\mathcal{T}$.

Proof.

Necessity. We use induction on $|V(G)|$. If $|V(G)|=6$, then $G=K_{3,3}\in\mathcal{T}$, and so the result holds. Assume that $|V(G)|\geq 8$ and the result holds for graphs with less vertices than $G$. If $G$ is 3-connected, then $|V(G)|<10$ by Lemma 4.6, and so $G=K_{3,3}\in\mathcal{T}$. Suppose that $G$ is not 3-connected. Then $G$ has a 2-cut. By Lemma 4.8(i), we can take a 2-cut $\partial{(X)}=\{uv,wz\}$ of $G$ such that $u,w\in X$, $G[X]+uw\in\mathcal{\widehat{H}}$ and $G[\overline{X}\cup\{u,w\}]+uw\in\mathcal{H}^{\diamond}$. Assume that $G[\overline{X}\cup\{u,w\}]+uw=(L_{n}\oplus H)_{e,f}$ for some $n\geq 1$, where $H\in\mathcal{H}$ and $e$ is a 22-edge of $L_{n}$. By Lemma 4.7(ii), each nice pair set $(A_{1},B_{1})$ (resp. $(A_{H},B_{H})$) of $G[X]+uw$ (resp. $H$) satisfies $|A_{1}|\leq 3$ and $|B_{1}|\leq 3$ (resp. $|A_{H}|\leq 3$ and $|B_{H}|\leq 3$). So $G[X]+uw\cong K_{3,3}$ by Lemma 4.6 and $H\in\mathcal{T}$ by the induction hypothesis. Since $G\cong(K_{3,3}\oplus(L_{n}\oplus H)_{e,f})_{e^{\prime}}=(H\oplus(L_{n}\oplus K_{3,3})_{e^{\prime}})_{f,e}$, we have that $G\in\mathcal{T}$, where $e^{\prime}$ is another 22-edge of $L_{n}$. So the necessity holds.

Sufficiency. The result is obvious if $G=K_{3,3}$. So suppose that $G\neq K_{3,3}$. Then $|V(G)|\geq 12$. By induction on $|V(G)|$. If $|V(G)|=12$, then $G=(K_{3,3}\oplus(L_{1}\oplus K_{3,3})_{e_{1}})_{e_{2}}$, where $\{e_{1},e_{2}\}$ is a matching of $L_{1}$. Since each nice pair set $(A^{*},B^{*})$ of $K_{3,3}$ satisfies $|A^{*}|\leq 3$ and $|B^{*}|\leq 3$, the result follows from Lemma 4.7. Assume that $|V(G)|\geq 14$ and the result holds for graphs with less vertices than $G$. Let $G=(T\oplus(L_{n^{\prime}}\oplus K_{3,3})_{e_{4},e_{3}})_{e_{1},e_{2}}$ (see Fig. 7), where $T\in\mathcal{T}$ and $e_{i}=u_{i}v_{i}$ for $1\leq i\leq 4$. Set $U=V(L_{n^{\prime}})\setminus\{u_{i},v_{i}|i=2,4\}$. By Lemma 4.7(i), $A^{\prime}\cup B^{\prime}$ is contained in $V(T)$, $V(K_{3,3})$ or $U$. Observe that for any $a\in U\cap A$ (resp. $b\in U\cap B$), there is exactly one vertex $b\in U\cap B$ (resp. $a\in U\cap A$) of $G$ such that $(a,b)$ is a nice pair of $G$. So $|A^{\prime}|=|B^{\prime}|=1$ if $A^{\prime}\cup B^{\prime}\subseteq U$. If $A^{\prime}\cup B^{\prime}\subseteq V(T)$ (or $A^{\prime}\cup B^{\prime}\subseteq V(K_{3,3})$), then $(A^{\prime},B^{\prime})$ is also a nice pair set of $T$ (or $K_{3,3}$) by Lemma 4.7(ii). If $T\neq K_{3,3}$, then $|A_{T}|\leq 3$ and $|B_{T}|\leq 3$ for each nice pair set $(A_{T},B_{T})$ of $T$ by induction. So $|A^{\prime}|\leq 3$ and $|B^{\prime}|\leq 3$, and the sufficiency is proved.

∎

Theorem 4.1 is obtained directly from Lemmas 4.6, 4.8(ii) and 4.9. So we prove Corollary 4.2.

Proof of Corollary 4.2. If $G$ is 3-connected, then $G$ has at least 9 nice pairs of vertices and equality holds if and only if $G=K_{3,3}$ by Lemma 4.6. If $G$ is not 3-connected, by Lemma 4.8(i), take a 2-cut $\partial{(X)}=\{uv,wz\}$ of $G$ such that $u,w\in X$, $G[X]+uw\in\mathcal{\widehat{H}}$ and $G[\overline{X}\cup\{u,w\}]+uw\in\mathcal{H}^{\diamond}$. Assume that $G[\overline{X}\cup\{u,w\}]+uw=(L_{n}\oplus H)_{e,f}$ for some $n\geq 1$, where $H\in\mathcal{H}$ and $e$ is a 22-edge of $L_{n}$. Then $G[X]+uw$ and $H$ both have at least 9 nice pairs of vertices by Theorem 4.1. Using Lemma 4.7(ii), $G$ has at least 18 nice pairs of vertices, and so the corollary holds. $\square$

5 Conclusion and further problems

In this paper, we characterized nice vertices in cubic graphs in terms of barriers (see Theorem 2.3), which was defined as trimatched vertices in [9]. By using some results of matching covered graphs, we mainly obtained two lower bounds on the number of nice vertices in non-bipartite cubic graphs and determined all the corresponding extremal graphs. We also defined nice pairs of vertices in cubic bipartite graphs and showed that a cubic bipartite graph is a brace if and only if each pair of vertices in different colors is nice (see Proposition 4.4). We also showed that every cubic bipartite graph has at least 9 nice pairs of vertices.

However, there are still some problems that may be further considered. Of course there is a polynomial algorithm to determine whether a vertex $u$ in a cubic graph $G$ is nice: First delete vertex $u$ together with three neighbors from $G$, then check whether the remained graph has a perfect matching or not by using known matching algorithm. Then $u$ is nice or not according to our answer “yes” or “no”. From this we can determine all nice vertices in cubic graphs. Problem 1. Is there a faster algorithm to determine all nice vertices in cubic graphs ?

Lemma 1.2 states that each vertex in a cubic bicritical graph is nice. But there exist some non-bicritical cubic graphs in which all vertices are nice. Let $G$ be a cubic graph obtained from a cubic brace $H(A,B)$ by splicing a cubic brick at each vertex $b\in B$. For example, see Fig. 3(c) for $H(A,B)=K_{3,3}$ and the cubic brick $K_{4}$. Then $G$ is a non-bicritical graph since the deletion of any two vertices in $A$ results in a graph with no perfect matching. But each vertex of $G$ in $A$ is nice by Lemma 3.2 and the other vertices of $G$ are also nice by Lemma 2.12. So we propose the following problem.

Problem 2. Characterize all cubic graphs whose all vertices are nice.

Finally, we can also generalize the concept of nice vertices in cubic graphs to general graphs as follows. A vertex $u$ of a graph $G$ is nice if $u$ and three of its neighbors induce a nice subgraph of $G$. Then we can consider some similar problems in regular graphs.

Problem 3. How many nice vertices do $k$-regular graphs of even order have for $k\geq 4$ ?