A 2-Bisection with Small Number of Monochromatic Edges of a Claw-Free Cubic Graph

Abstract

A bisection of a graph G is a partition of its vertex set into two parts of the same cardinality. A k-bisection of G is a bisection of G such that every component of each part has at most k vertices. Cui and Liu proved that every claw-free cubic graph contains a 2-bisection. In this paper, we improve this result showing that every claw-free cubic graph contains a 2-bisection with bounded number of monochromatic edges, where a monochromatic edge of a 2-bisection is an edge connecting two vertices of the same part of the 2-bisection. We also prove that our bound is best possible for all claw-free cubic simple graphs.

1 Introduction

In this paper, we consider graphs that possibly contain multiple edges, but do not allow loops. A graph without multiple edges is said to be simple. A bisection of a graph G is a partition (B, W) of V(G) so that $|B| = |W|$. A k-bisection of a graph G is a bisection (B, W) of G such that every component of G[B] and G[W] has at most k vertices, where G[S] denotes the subgraph of G induced by S for a vertex set S of G. Note that a k-bisection of a graph is an m-bisection for $m \ge k$.

As explained by Esperet et al. [5], a result due to Jaeger [6] shows that a cubic graph with a circular nowhere-zero r-flow contains an $(\lfloor r \rfloor -2)$-bisection. With this relation in mind, many researchers have studied a k-bisection of a cubic graph. The following is an important conjecture.

Conjecture 1

(Ban and Linial [3]) Every bridgeless cubic simple graph contains a 2-bisection, except for the Petersen graph.

Esperet et al. [5] gave an infinite family of cubic simple graphs G with bridges such that G contains no 2-bisection. They also proved that every cubic simple graph contains a 3-bisection. As a positive evidence for Conjecture 1, Abreu et al. [2] confirmed Conjecture 1 for cycle permutation cubic graphs.

A claw is the complete bipartite graph $K_{1,3}$. A graph that contains no induced claw is said to be claw-free. Abreu et al. [1] proved that every bridgeless claw-free cubic simple graph contains a 2-bisection, which gives a partial solution of Conjecture 1. Later, Cui and Liu [4] pointed out that the result holds without assuming bridgelessness and simpleness.

Theorem 1.1

(Cui and Liu [4]) Every claw-free cubic graph contains a 2-bisection.

Note that a cubic graph G contains a 1-bisection if and only if G is bipartite. In this sense, the smaller integer k for which a cubic graph contains a k-bisection, the closer to being bipartite it is. However, two 2-bisections of the same cubic graph may not have the same meaning in the sense of how close to being a bipartite graph. To consider this, we focus on the number of monochromatic edges in a 2-bisection, where monochromatic edges are ones connecting two vertices of the same part of the 2-bisection.

For example, consider the two 2-bisections (B, W) on the same graph in Fig. 1. (In this paper, we always mean that B is the set of black vertices and W is the set of white vertices.) The 2-bisection in the left contains six monochromatic edges, while the 2-bisection in the right contains only four monochromatic edges. Thus, the 2-bisection in the right contains edges connecting different parts more than one in the left. We can consider that the 2-bisection in the right asserts the property that the graph is close to being bipartite more strongly than the one in the left.

Fig. 1

2-bisections (B, W) on the same graph, where B is the set of black vertices and W is the set of white vertices

As in these examples, the less number of monochromatic edges a 2-bisection of a cubic graph G has, the closer to being bipartite it shows that G is. With this thought in mind, in this paper, we consider the number of monochromatic edges in a 2-bisection of a claw-free cubic graph.

We denote by $\varepsilon (B, W)$ the number of monochromatic edges in a 2-bisection (B, W). A diamond in a graph is an induced subgraph isomorphic to the complete graph $K_4$ with one edge deleted. It is easy to see that any two diamonds in a connected claw-free cubic graph G are vertex-disjoint, unless G is $K_4$.

The following is our main theorem, which shows that every claw-free cubic graph contains a 2-bisection with bounded number of monochromatic edges. This is an improvement of Theorem 1.1.

Theorem 1.2

Let G be a connected claw-free cubic graph that is not $K_4$, and let k be the number of diamonds in G. Then G contains a 2-bisection (B, W) such that

$$\begin{aligned} \varepsilon (B, W) \le {\left\{ \begin{array}{ll} \dfrac{|V(G)|-k}{3} &{} \text {if}\,\, k \,\, \text {is even},\\ \dfrac{|V(G)|-k}{3} +1 &{} \text {if}\,\, k \,\, \text {is odd}. \end{array}\right. } \end{aligned}$$

We would like to emphasize that the bound in Theorem 1.2 is best possible for all connected claw-free cubic simple graphs other than $K_4$. In fact, the following proposition holds.

Proposition 1.3

Let (B, W) be a 2-bisection of a connected claw-free cubic simple graph G that is not $K_4$, and let k be the number of diamonds in G. Then

$$\begin{aligned} \varepsilon (B, W) \ge {\left\{ \begin{array}{ll} \dfrac{|V(G)|-k}{3} &{} \text {if} \,\,\, k \,\,\, \text {is even},\\ \dfrac{|V(G)|-k}{3} +1 &{} \text {if}\,\,\, k \, \,\,\text { is odd}. \end{array}\right. } \end{aligned}$$

In Sect. 2, we prove Proposition 1.3, together with some preliminaries.

We prove Theorem 1.2 in Sect. 3. The proof is based on combination of the ideas in Ref. [4] and in Ref. [1]: First we reduce the problem to the case of bridgeless claw-free cubic simple graphs in Sects. 3.2 and 3.3, using some reductions similar to the ones in Ref. [4]. In Sect. 3.4, we use a characterization of a bridgeless claw-free cubic simple graphs due to Oum [7], and then consider a 2-factor in a certain cubic graph. We follow those ideas in Ref. [4] and in Ref. [1], but in order to deal with monochromatic edges, our proof requires essentially new arguments. In fact, some reductions in Ref. [4] does not work in our purpose, and hence we introduce new reductions in this paper.

2 Properties on the Number of Monochromatic Edges

We give some basic definitions and set up terminology to prove Theorem 1.2.

For a vertex set S of a graph G, we denote by G[S] the subgraph of G induced by S. For an edge set T in G, we denote the set of end vertices of edges in T by V(T). When T is singleton, i.e. $T = \{e\}$, then we simply write V(e) for $V(\{e\})$. Recall that an edge of a connected graph is called a bridge, if its deletion results in a disconnected graph. A connected graph without bridges is said to be bridgeless. An edge set T of a connected graph G is an edge-cut if $G - T$ is disconnected.

In the proof of Theorem 1.2, we use the following theorem due to Petersen [8].

Theorem 2.1

Every bridgeless cubic graph contains a perfect matching.

We need a characterization of bridgeless claw-free cubic simple graphs, by employing the notations in Ref. [7]. A string of diamonds is a maximal sequence $A_{1}, A_{2}, \dots , A_{k}$ of diamonds, where $A_i$ is a diamond for each $i \in \lbrace 1, 2, \dots , k \rbrace$, and $A_{i}$ has a vertex adjacent to a vertex in $A_{i+1}$ for $i \in \{1,2, \dots , k-1\}$. A string of diamonds has exactly two vertices of degree 2, which are called the head and the tail of the string. For an edge $e = uv$ in a cubic graph H, replacing e with a string of diamonds with the head x and the tail y is to remove e and add the string of diamonds together with edges ux and vy.

For a vertex v in a cubic graph H, replacing v with a triangle is to replace v with three new vertices $v_1, v_2, v_3$ forming a triangle so that if $u_1, u_2, u_3$ are the neighbors of v in H, then $v_i$ and $u_i$ are adjacent in the new graph for $i \in \{1,2,3\}$.

A connected claw-free cubic graph in which every vertex is in a diamond is called a ring of diamonds. Now, we can describe the following structure of bridgeless claw-free cubic simple graphs.

Theorem 2.2

(Oum [7]) A bridgeless simple graph G is claw-free and cubic, if and only if either

• (1) G is isomorphic to $K_{4}$

• (2) G is a ring of diamonds, or

• (3) G can be built from a bridgeless cubic multigraph H by replacing some edges of H with strings of diamonds and replacing each vertex of H with a triangle (see Fig. 2).

Fig. 2

An example of a bridgeless claw-free cubic simple graph. Its vertex set can be partitioned into pairwise vertex-disjoint two diamonds and eight triangles

We now consider the number of monochromatic edges in a 2-bisection of a connected claw-free cubic graph G. The following lemma is basic, but important when we consider the number of monochromatic edges.

Lemma 2.3

Let (B, W) be a 2-bisection of a cubic graph G and let $\varepsilon _{B}$ (resp. $\varepsilon _{W}$) be the number of monochromatic edges in G[B] (resp. G[W]). Then $\varepsilon _{B} = \varepsilon _{W}$.

Proof

We count e(B, W), which is the number of edges between B and W. Considering the number of edges leaving from vertices in B, and also those from vertices in W, we have

$$\begin{aligned} e(B, W)= & {} 3 \vert B \vert - 2\varepsilon _{B} \ = \ 3 \vert W \vert - 2\varepsilon _{W}. \end{aligned}$$

Since $\vert B \vert = \vert W \vert$, we conclude $\varepsilon _{B}=\varepsilon _{W}$. $\square$

Using Lemma 2.3, we prove Proposition 1.3 as follows.

Proof of Proposition 1.3

Let (B, W) be a 2-bisection of a connected claw-free cubic simple graph G that is not $K_4$, and let k be the number of diamonds in G. Each triangle of G contains at least one monochromatic edge. Since G is a connected claw-free cubic simple graph that is not $K_4$, the vertex set V(G) can be partitioned into vertex-disjoint diamonds and triangles (see Fig. 2). Thus, it is easy to see that

$$\begin{aligned} \varepsilon (B, W) \ge k+ {{|V |- 4k} \over 3 } = {{|V |- k} \over 3 } . \end{aligned}$$

This completes the proof of the case when k is even.

Suppose that k is odd and $\varepsilon (B, W) = {{|V |- k} \over 3 }$. By Lemma 2.3, $\varepsilon (B, W) = \varepsilon _B + \varepsilon _W$ must be an even integer, where $\varepsilon _B$ (resp. $\varepsilon _W$) is the number of monochromatic edges in G[B] (resp. G[W]). However, this contradicts the fact that $\vert V \vert$ is even and k is odd. Thus, when k is odd, we have $\varepsilon (B, W) > {{|V |- k} \over 3 }$. Since the vertex set V(G) can be partitioned into pairwise vertex-disjoint diamonds and triangles, ${{|V |- k} \over 3 } = k + {{|V |- 4k} \over 3 }$ is an integer. Thus, we obtain $\varepsilon (B, W) \ge {{|V |- k} \over 3 } +1$, which completes the proof of Proposition 1.3. $\square$

We need the following lemma to reduce a connected claw-free cubic graph to smaller ones.

Lemma 2.4

Let G be a connected claw-free cubic graph that is not $K_4$. Suppose that G contains an edge-cut $\{u_1u_2,v_1v_2\}$ such that $u_2$ and $v_2$ belong to the same component of $G - \{u_1u_2,v_1v_2\}$ and $u_1 \ne v_1$. Then, the following hold.

• (1) Consider the graph obtained from G by deleting the edges $u_1u_2$ and $v_1v_2$, and adding a new edge connecting $u_1$ and $v_1$. Let $G_1$ be the component of the graph containing $u_1$ and $v_1$ (see the center of Fig. 3). Then $G_1$ is a connected claw-free cubic graph.

• (2) If $G_1$ is not $K_4$, then the new edge $u_1v_1$ forms multiple edges in $G_1$ or is contained in no triangle of $G_1$. In particular, the new edge $u_1v_1$ is contained in no diamond of $G_1$.

• (3) Let $G_1'$ be the graph obtained from $G_1$ by deleting the new edge $u_1v_1$ and adding four new vertices a, b, c, d and seven new edges $u_1a, ab, ac, bc, bd, cd, dv_1$ (see the right of Fig. 3). Then $G_1'$ is a connected claw-free cubic graph that is not $K_4$.

Fig. 3

The graphs $G_1$ and $G_1'$ in Lemma 2.4

Proof

• (1) It is trivial that $G_1$ is a connected cubic graph. Suppose that $G_1$ contains an induced claw. Since G is claw-free, it must use the new edge $u_1v_1$, which implies that the center is either $u_1$ or $v_1$. By symmetry, we may assume that its center is $u_1$. Then, using $u_2$ instead of $v_1$, we obtain an induced claw in G, a contradiction. Thus, $G_1$ is claw-free.

• (2) Suppose that the edge $u_1v_1$ does not form multiple edges in $G_1$, that is $u_1v_1 \notin E(G)$, and the new edge $u_1v_1$ in $G_1$ is contained in a triangle. Let x be the third vertex of the triangle. We may assume that there are no multiple edges in $G_1$ connecting $u_1$ and x, since otherwise, there are no multiple edges in $G_1$ connecting $v_1$ and x, and so we can exchange the roles of $u_1$ and $v_1$. Let y be the neighbor of $u_1$ in $G_1$ other than $v_1,x$. To forbid an induced claw with center $u_1$ in G, we have $xy \in E(G)$. Note that the neighbors of x are $u_1,v_1$ and y. If $v_1y \notin E(G)$, then there is an induced claw with center $v_1$ in G, a contradiction. Thus, we have $v_1y \in E(G)$, which implies that $G_1$ is $K_4$, a contradiction again. This shows that the new edge $u_1v_1$ forms multiple edges in $G_1$ or is contained in no triangle of $G_1$. In addition, this directly implies that the new edge is contained in no diamond of $G'$.

• (3) This can be proven similarly to (1).

$\square$

3 Proof of Theorem 1.2

3.1 A Desired Bisection

For a technical reason, we will find a bisection with particular conditions. For the ease of notation, we use the following terminology: Let G be a connected claw-free cubic graph that is not $K_4$. We call a bisection (B, W) of G a desired bisection, if it satisfies the following four conditions.

• (B1) For each triangle of G, each part of the bisection contains at least one of the three vertices. Thus, exactly one edge in the triangle is monochromatic, and none of the other two edges are monochromatic.

• (B2) Every monochromatic edge is contained in a triangle.

• (B3) Every diamond contains exactly one monochromatic edge.

• (B4) For two vertices u, v joining more than one edge (i.e. multiple edges), the edges between u and v are not monochromatic.

Lemma 3.1

A desired bisection in a connected claw-free cubic graph is a 2-bisection.

Proof

Let (B, W) be a desired bisection in a claw-free cubic graph G. Suppose that G contains a path P consisting of three vertices and two monochromatic edges. By Conditions (B1) and (B2), the two monochromatic edges in P are contained in distinct triangles of G. Since G is cubic, the triangles must share an edge, and hence they form a diamond. However, by Condition (B3), the diamond cannot contain the two edges of P, a contradiction. Thus, such a path P does not exist, which directly implies that (B, W) is a 2-bisection of G. $\square$

We prove the following theorem, which implies Theorem 1.2 as follows.

Theorem 3.2

Let G be a connected claw-free cubic graph that is not $K_4$. If the number of diamonds in G is even, then G contains a desired bisection.

Proof of Theorem 1.2 assuming Theorem 3.2

Assume that Theorem 3.2 holds. Let G be a connected claw-free cubic graph that is not $K_4$, and let k be the number of diamonds.

If k is even, then it follows from Theorem 3.2 that G contains a desired bisection (B, W). By Lemma 3.1, (B, W) is a 2-bisection. Since any two diamonds are vertex-disjoint, it follows from Conditions (B1), (B2) and (B3) that

$$\begin{aligned} \varepsilon (B, W) \le k + \dfrac{|V(G)|-4k}{3} = \dfrac{|V(G)|-k}{3}, \end{aligned}$$

as required.

Thus, we may assume that k is odd. In this case, there exists a diamond in G. Let a, b, c, d be the four vertices of the diamond with $ad \notin E(G)$, and let x (resp. y) be the neighbor of a (resp. d) other than b, c. Note that $\{xa, yd\}$ is an edge-cut of G. Let $G'$ be the graph obtained from G by deleting the vertices a, b, c, d and adding a new edge connecting x and y (see Fig. 4). If $xy \in E(G)$, then the new edge, together with the edge in G, forms multiple edges connecting x and y. By Lemma 2.4 (1), $G'$ is a connected claw-free cubic graph.

If $G'$ is $K_4$, then G has exactly eight vertices and G is a ring of diamonds with two diamonds, which contradicts the fact that k is odd. Thus, $G'$ is not $K_4$. By Lemma 2.4 (2), there is no diamond containing the edge xy, and hence the number of diamonds in $G'$ is $k-1$, which is even. By Theorem 3.2, there is a desired bisection $(B', W')$ in $G'$. By Lemma 3.1, $(B',W')$ is a 2-bisection. In particular, as explained above, we have

$$\begin{aligned} \varepsilon (B', W') \le \dfrac{|V(G')|-(k-1)}{3} = \dfrac{(|V(G)|-4)-(k-1)}{3} = \dfrac{|V(G)|-k}{3}-1. \end{aligned}$$

By Lemma 2.4 (2), the new edge xy in $G'$ forms multiple edges in $G'$ or is contained in no triangle of $G'$. By Conditions (B2) and (B4), we see that the new edge xy is not monochromatic. By the symmetry between x and y, we may assume that $x \in B'$ and $y \in W'$ (see the right of Fig. 4). Let $B = B' \cup \{b,d\}$ and $W = W' \cup \{a,c\}$ (see the left of Fig. 4). Then, we have

$$\begin{aligned} \varepsilon (B, W) = \varepsilon (B', W') + 2 \le \dfrac{|V(G)|-k}{3}+1, \end{aligned}$$

as desired. This completes the proof of Theorem 1.2. $\square$

Fig. 4

The 2-bisections (B, W) of G and $(B',W')$ of $G'$ in the case k is odd in the proof of Theorem 1.2

3.2 Preliminaries for the Proof

As in the previous subsection, it suffices to prove Theorem 3.2. The proof consists of three steps. First, we deal with two preliminary results, one of which concerns multiple edges and the other concerns two adjacent diamonds. The remaining proof is divided into two cases depending on whether G contains a bridge.

In the remaining of this section, we assume that Theorem 3.2 does not holds. Under the assumption, a minimum counterexample means a minimum graph among all connected claw-free cubic graphs G that is not $K_4$ with even number of diamonds such that G contains no desired bisection.

Lemma 3.3

No minimum counterexample contains multiple edges connecting two vertices, say u and v, such that $x \ne y$, where x is the unique neighbor of u other than v and y is the unique neighbor of v other than u.

Proof

Assume that such multiple edges exist in a minimum counterexample G. Let $G'$ be the graph obtained from G by deleting u and v and adding a new edge connecting x and y. (see Fig. 5). Since $\lbrace ux, vy \rbrace$ is an edge-cut of G, it follows from Lemma 2.4 (1) that $G'$ is a connected claw-free cubic graph. If $G'$ is $K_4$, then G has exactly six vertices and there is only one diamond, contradicting the assumption that the number of diamonds is even. Thus, $G'$ is not $K_4$. By Lemma 2.4 (2), there is no diamond containing the edge xy, and hence the number of diamonds in $G'$ is the same as that of G. Since G is a minimum counterexample, $G'$ contains a desired bisection $(B', W')$.

By Lemma 2.4 (2), the new edge xy forms multiple edges in $G'$ or is contained in no triangle of $G'$. Thus, by Conditions (B2) and (B4) for $(B', W')$, the edge xy is not monochromatic. By the symmetry between x and y, we may assume that $x \in B'$ and $y \in W'$ (see the right of Fig. 5). Let $B = B' \cup \{v\}$ and $W = W' \cup \{u\}$ (see the left of Fig. 5). Since $(B', W')$ is a desired bisection of $G'$, it is easy to see that (B, W) is a desired bisection of G, contradicting the fact that G is a counterexample. $\square$

Fig. 5

The 2-bisection (B, W) of G and $(B', W')$ of $G'$ in the proof of Lemma 3.3

Lemma 3.4

No minimum counterexample contains two diamonds that are adjacent each other.

Proof

Assume that such two diamonds exist in a minimum counterexample G. Let $a_1,b_1,c_1,d_1, a_2,b_2,c_2,d_2$ be the vertices such that $\{a_i,b_i,c_i,d_i\}$ forms a diamond for $i=1,2$, $d_1 a_2 \in E(G)$ and $a_1d_1, a_2d_2 \notin E(G)$.

If $a_1d_2 \in E(G)$, then G is a ring of diamonds consisting of two diamonds. In this case, (B, W), where $B = \{b_1, c_1, a_2, d_2\}$ and $W = \{a_1, d_1, b_2, c_2\}$, is clearly a desired bisection, a contradiction.

Let x (resp. y) be the neighbor of $a_1$ (resp. $d_2$) other than $b_1,c_1$ (resp. $b_2,c_2$). If $x=y$, there is an induced claw with center x, a contradiction. Thus $x \ne y$. Let $G'$ be the graph obtained from G by deleting $a_1,b_1,c_1,d_1,a_2,b_2,c_2$ and $d_2$, and adding a new edge connecting x and y (see Fig. 6). By Lemma 2.4 (1), G is a connected claw-free cubic graph. If we let k be the number of diamonds in G, then $G'$ contains $k-2$ diamonds. Since k is even, so is $k-2$.

If $G'$ is $K_4$, then G is a ring of diamonds consisting of three diamonds, which contradicts the assumption that k is even. Thus, we may assume that $G'$ is not $K_4$. Since G is a minimum counterexample, $G'$ contains a desired bisection $(B', W')$. By Lemma 2.4 (2), the new edge xy in $G'$ forms multiple edges in $G'$ or is contained in no triangle of $G'$. Thus, by Conditions (B2) and (B4) for $(B', W')$, the edge xy is not monochromatic. By the symmetry between x and y, we may assume that $x \in B'$ and $y \in W'$ (see the right of Fig. 6). Let $B = B' \cup \{b_1, c_1, a_2, d_2\}$ and $W = W' \cup \{a_1, d_1, b_2, c_2\}$, (see the left of Fig. 6). Then, it is easy to see that (B, W) is a desired bisection, a contradiction. $\square$

Fig. 6

The 2-bisection (B, W) of G and $(B', W')$ of $G'$ in the proof of Lemma 3.4

3.3 The Case When G Contains a Bridge

In this subsection, we show that no minimum counterexample contains a bridge.

Lemma 3.5

No minimum counterexample contains a bridge.

Proof

Let G be a minimum counterexample, and let k be the number of diamonds in G. Assume that G contains a bridge $e = v_1v_2$ and we label vertices nearby e as in Fig. 7. Note that $x_1 \ne y_1$ and $x_2 \ne y_2$ by Lemma 3.3. We divide the remaining proof into three cases, depending on whether $z_1 = w_1$ and $z_2 = w_2$.

Fig. 7

A label of vertices nearby the bridge e in the proof of Lemma 3.5

Case 1: $z_1 = w_1$ and $z_2=w_2$.

In this case, $\{z_1,x_1,y_1,v_1\}$ and $\{v_2,x_2,y_2,z_2\}$ form diamonds that are adjacent each other. However, this contradicts Lemma 3.4.

Fig. 8

Case 2 in the proof of Lemma 3.5

Case 2: Either $z_1 = w_1$ and $z_2 \ne w_2$, or $z_1 \ne w_1$ and $z_2=w_2$.

By symmetry, we may assume that $z_1 = w_1$ and $z_2 \ne w_2$. Note that $z_1=w_1$ is incident with another bridge $e^{\prime }$. Let $v_3$ be the other end vertex of $e^{\prime }$, and label the vertices around $v_3$ as shown in the left of Fig. 8. Note that $x_3 \ne y_3$ by Lemma 3.3. If $z_3=w_3$, then $\{z_1,x_1,y_1,v_1\}$ and $\{v_3,x_3,y_3,z_3\}$ form diamonds that are adjacent each other, contradicting Lemma 3.4. Thus, we may assume $z_3 \ne w_3$.

Consider the graph obtained from G by deleting $v_1,x_1,y_1,z_1,v_2, x_2,y_2,v_3, x_3$ and $y_3$ and adding a new edge connecting $z_2$ and $w_2$, and another new edge connecting $z_3$ and $w_3$. It has two components, one of which contains the new edge $z_2w_2$, and the other contains $z_3w_3$. Let $G_2$ be the former one, and $G_3$ be the latter one. By Lemma 2.4 (1), both of $G_2$ and $G_3$ are connected claw-free cubic graphs. Note that $G_2$ and $G_3$ are symmetric.

For $i=2,3$, let $k_i$ be the number of diamonds in G using only the vertices in $G_i$. Note that $k=k_2+k_3+1$. Since k is even, we may assume that $k_2$ is even and $k_3$ is odd by symmetry.

If $G_2$ is $K_4$, then $k_2 = 1$ a contradiction. Thus, $G_2$ is not $K_4$. Since G is a minimum counterexample, $G_2$ contains a desired bisection $(B_2, W_2)$. By Lemma 2.4 (2), the new edge $z_2w_2$ forms multiple edges in $G_2$ or is contained in no triangle of $G_2$, and hence it is not a monochromatic edge by Conditions (B2) and (B4). By symmetry, we may assume $z_2 \in B_2$ and $w_2 \in W_2$.

Let $G_3'$ be the graph obtained from $G_3$ by deleting the new edge $z_3w_3$ and adding four new vertices a, b, c, d and seven new edges $z_3a, ab, ac, bc, bd, cd, dw_3$ (see the right of Fig. 8). By Lemma 2.4 (3), $G_3'$ is a connected claw-free cubic graph such that $G_3'$ is not $K_4$ and $|V(G_3')| < |V(G)|$. The number of diamonds in $G_3'$ is $k_3+1$, which is even since $k_3$ is odd. Since G is a minimum counterexample, $G_3'$ contains a desired bisection $(B_3',W_3')$. By Condition (B3), the edge bc must be monochromatic. By symmetry, we may assume $b,c \in B_3'$, which implies that $a,d \in W_3'$ and $z_3,w_3 \in B_3'$ by Conditions (B2) and (B3).

Let

$$\begin{aligned} B= & {} B_2 \cup \big (B_3'\setminus \{b,c\}\big ) \cup \{v_3, x_1, y_1, v_2, y_2\},\\ W= & {} W_2 \cup \big (W_3'\setminus \{a,d\}\big ) \cup \{x_3, y_3, z_1, v_1, x_2\} \end{aligned}$$

(see the left of Fig. 8). Since $(B_2, W_2)$ is a desired bisection of $G_2$ and $(B_3', W_3')$ is a desired bisection of $G_3'$, it is easy to see that (B, W) is a desired bisection of G, a contradiction. This completes the proof of Case 2.

Case 3: $z_1 \ne w_1$ and $z_2 \ne w_2$,

In this case, consider the graph obtained from G by deleting $v_1,x_1,y_1,v_2, x_2,y_2$ and adding a new edge connecting $z_1$ and $w_1$, and another new edge connecting $z_2$ and $w_2$. It has two components, one of which contains the new edge $z_1w_1$, and the other contains $z_2w_2$. Let $G_1$ be the former one, and $G_2$ be the latter one (see Fig. 9). By Lemma 2.4 (1), both $G_1$ and $G_2$ are connected claw-free cubic graphs.

For $i=1,2$, let $k_i$ be the number of diamonds in G using only the vertices in $G_i$. Note that $k=k_1+k_2$. Since k is even, either (i) both $k_1$ and $k_2$ are even or (ii) both $k_1$ and $k_2$ are odd. We consider these two cases individually.

Case 3-(i): Both $k_1$ and $k_2$ are even.

In this case, we see that neither $G_1$ nor $G_2$ is $K_4$. Since G is a minimum counterexample, for $i=1,2$, the graph $G_i$ contains a desired bisection $(B_i, W_i)$. By Lemma 2.4 (2), the new edge $z_iw_i$ forms multiple edges in $G_i$ or is contained in no triangle of $G_i$, and hence it is not a monochromatic edge by Conditions (B2) and (B4) (see the right of Fig. 9). By symmetry, we may assume $z_i \in B_i$ and $w_i \in W_i$. Let

$$\begin{aligned} B= & {} B_1 \cup B_2 \cup \{v_1, y_1, y_2\},\\ W= & {} W_1 \cup W_2 \cup \{x_1, v_2, x_2\} \end{aligned}$$

(see the left of Fig. 9). Since $(B_i, W_i)$ is a desired bisection of $G_i$ for $i=1,2$, we see that (B, W) is a desired bisection of G, a contradiction. This completes the proof of Case 3-(i).

Fig. 9

Case 3-(i) in the proof of Lemma 3.5

Case 3-(ii): Both $k_1$ and $k_2$ are odd.

For $i=1,2$, let $G_i'$ be the graph obtained from $G_i$ by deleting the edge $z_iw_i$ by adding four new vertices $a_i,b_i,c_i,d_i$ and seven new edges $z_ia_i, a_ib_i, a_ic_i, b_ic_i, b_id_i, c_id_i, d_iw_i$. By Lemma 2.4 (3), $G_i'$ is a connected claw-free cubic graph such that $G_i'$ is not $K_4$ and $|V(G_i')| < |V(G)|$. The number of diamonds in $G_i'$ is $k_i+1$, which is even since $k_i$ is odd. Since G is a minimum counterexample, $G_i'$ contains a desired 2-bisection $(B_i',W_i')$. By Condition (B3), the edge $b_ic_i$ must be monochromatic. By symmetry, we may assume $b_i,c_i \in B_i'$, which implies that $a_i,d_i \in W_i'$ and $z_i,w_i \in B_i'$ by Conditions (B2) and (B3). Let

$$\begin{aligned} B= & {} \big (B_1'\setminus \{b_1,c_1\}\big ) \cup \big (W_2'\setminus \{a_2,d_2\}\big ) \cup \{v_1, x_2, y_2\},\\ W= & {} \big (W_1'\setminus \{a_1,d_1\}\big ) \cup \big (B_2'\setminus \{b_2,c_2\}\big ) \cup \{v_2, x_1, y_1\} \end{aligned}$$

(see the left of Fig. 10). Since $(B_i', W_i')$ is a desired bisection of $G_i$ for $i=1,2$, we see that (B, W) is a desired bisection of G, a contradiction. This completes the proof of Case 3. $\square$

Fig. 10

Case 3-(ii) in the proof of Lemma 3.5

3.4 The Proof of Theorem 3.2

Now, we are ready to prove Theorem 3.2.

Proof of Theorem 3.2

Assume that Theorem 3.2 does not hold, and let G be a minimum counterexample. By Lemma 3.5, G is bridgeless. Suppose that G contains multiple edges connecting two vertices, say u and v. If u and v are joined by the third edge, then $(\{u\},\{v\})$ is trivially a desired bisection, a contradiction. Thus, it follows from Lemma 3.3 that $x=y$, where x is the unique neighbor of u other than v and y is the unique neighbor of v other than u. Then the edge incident with x other than ux, vx must be a bridge of G, contradicting the fact that G is bridgeless. Therefore, G is a simple graph.

By Lemma 3.4, G is not a ring of diamonds. Thus, by Theorem 2.2, G can be built from a bridgeless cubic multigraph $H'$ by replacing some edges of $H'$ with strings of diamonds and replacing each vertex of $H'$ with a triangle. Note that each string of diamonds contains only one diamond by Lemma 3.4. Let H be the graph obtained from $H'$ by replacing each vertex of $H'$ with a triangle, (see Fig. 11). Note that

• G is obtained from H by replacing some edges of H with strings of diamonds,

• H consists of pairwise vertex-disjoint $|V(H')|$ triangles.

• Each edge in $H'$ naturally corresponds to an edge connecting two triangles in H. Thus, with abuse of notation, we regard $E(H')$ as a subset of E(H). In particular, $E(H')$ is a perfect matching in H.

We call an edge e of H a substitution edge if e is replaced with a string of diamonds when we obtain G from H. The number of substitution edges in H is equal to the number of diamonds in G, which is even. The following claim is essential in our proof:

Fig. 11

The graphs H and $H'$ obtained from the connected claw-free cubic simple graph G in Fig. 2. In H, double lines, bold lines and thin lines represent substitution edges, edges in $F_1$ and those in $F_2$, respectively. (The edge represented by upper double lines is contained also in $F_1$.) In $H'$, the bold lines represent the 2-factor $F_1$ in $H'$

Claim 1

H contains a bisection $(B_H, W_H)$ satisfying the following conditions:

$({\hbox {B1}}')$ :

For each triangle of H, each part of the bisection contains at least one of the three vertices. Thus, exactly one edge in the triangle is monochromatic, and none of the other two edges are monochromatic.

$({\hbox {B2}}')$ :

Every monochromatic edge is either contained in a triangle or a substitution edge.

$({\hbox {B3}}')$ :

Each substitution edge is monochromatic.

Note that Condition (B1$'$) is the same as Condition (B1). Condition (B2$'$) corresponds to Condition (B2), but adapted to deal with substitution edges.

Proof of Claim 1

By Theorem 2.1, $H'$ contains a perfect matching M, and let $F_1$ be a 2-factor of $H'$ with $F_1 = E(H') \setminus M$. Define

$$\begin{aligned} F_{2}= \lbrace xy \in E(H)\setminus F_1 \, \vert \, \text{ both } \, x \, \text{ and } \, y \, \text{ are } \text{ incident } \text{ to } \text{ edges } \text{ of } \, F_1 \rbrace , \end{aligned}$$

(see the left of Fig. 11). It is easy to see the following:

• Each edge in $F_2$ is contained in a triangle of H.

• Each triangle of H contains an edge in $F_2$ and a vertex that is not an end vertex of the edge. The third vertex is incident to an edge in M, but to neither edges in $F_1$ nor those in $F_2$.

Note that $F_{1} \cup F_{2}$ induces pairwise vertex-disjoint cycles of H, and let ${\mathcal {F}}$ be the set of such cycles. Note that $\bigcup _{C \in {\mathcal {F}}} V(C) \cup V(M) = V(H)$ and $V(C) \cap V(M) = \emptyset$ for each $C \in {\mathcal {F}}$. Since each cycle in ${\mathcal {F}}$ passes through edges in $F_{1}$ and in $F_{2}$ alternatively, it is an even cycle. We call a cycle in ${\mathcal {F}}$ an E-cycle (resp. an O-cycle) if it contains an even (resp. odd) number of substitution edges. Since the number of substitution edges is even, the sum of the number of O-cycles and the number of substitution edges in M is even.

For each O-cycle C in ${\mathcal {F}}$, choose an edge $e_C$ in M such that an end vertex of $e_C$ and the end vertices of some edge in $E(C) \cap F_2$ form a triangle of H. If $e_C = e_{C'}$ for some O-cycle $C'$ with $C' \ne C$, then we call C and $C'$ a friendly pair. If $e_C \ne e_{C'}$ for any O-cycle $C'$ with $C' \ne C$, then we say that C is isolated. We define the following sets:

$$\begin{aligned} {\mathcal {F}}_{1}= & {} \lbrace C \in {\mathcal {F}} \, \vert \, C \,\text{ is } \text{ an } \, E\text{-cycle } \rbrace \\ {\mathcal {F}}_{2}= & {} {} \lbrace H[V(C) \cup V(e_{C})] \, \vert \, C \in {\mathcal {F}} \, \text{ is } \text{ an } \text{ isolated } \, O\text{-cycle } \text{ and } \, e_{C} \, \text{ is } \text{ not } \text{ a } \text{ substitution } \text{ edge } \rbrace \\ {\mathcal {F}}_{3}= & {} \lbrace H[V(C) \cup V(e_{C})] \, \vert \, C \in {\mathcal {F}} \, \text{ is } \text{ an } \text{ isolated } \, O\text{-cycle } \text{ and }\, e_{C} \, \text{ is } \text{ a } \text{ substitution } \text{ edge } \rbrace \\ {\mathcal {F}}_{4}= & {} {} \lbrace H[V(C) \cup V(C') \cup V(e_{C})] \, \vert \, C,C' \in {\mathcal {F}} \, \text{ form } \text{ a } \text{ friendly } \text{ pair, } \\{} & {} \text{ and }\, e_{C} \, \text{ is } \text{ not } \text{ a } \text{ substitution } \text{ edge } \rbrace \\ {\mathcal {F}}_{5}= & {} {} \lbrace H[V(C) \cup V(C') \cup V(e_{C})] \, \vert \, C,C' \in {\mathcal {F}}\, \text{ form } \text{ a } \text{ friendly } \text{ pair, }\\{} & {} {}{} \text{ and } \, e_{C} \, \text{ is } \text{ a } \text{ substitution } \text{ edge } \rbrace \\ {\mathcal {F}}_{6}= & {} \lbrace H[v(e)] \, \vert \, e \in M, \, e \ne e_{C} \text{ for } \text{ any } \, O \text{-cycle }\, C \in {\mathcal {F}}\, \text{ and } \, e \, \text{ is } \text{ not } \text{ a } \text{ substitution } \text{ edge } \rbrace \\ {\mathcal {F}}_{7}= & {} \lbrace H[v(e)] \, \vert \, e \in M, \, e \ne e_{C} \text{ for } \text{ any }\, O \text{-cycle }\, C \in {\mathcal {F}}\, \text{ and } \, e \, \text{ is } \text{ a } \text{ substitution } \text{ edge } \rbrace \end{aligned}$$

By the construction, $V(P) \cap V(P') = \emptyset$ for any $P,P' \in \bigcup _{i=1}^7 {\mathcal {F}}_i$ with $P \ne P'$. Since every vertex in H is either contained in some $C \in {\mathcal {F}}$ or incident to an edge in M, we see

$$\begin{aligned} V(H) = \bigcup _{i=1}^{7}{\bigcup _{P \in {\mathcal {F}}_{i}}{V(P)}}. \end{aligned}$$

For each $P \in \bigcup _{i=1}^7 {\mathcal {F}}_i$, we denote by $s_P$ the number of substitution edges contained in P. By the definition, $s_P$ is even for $P \in {\mathcal {F}}_i$ if and only if $i = 1, 3,4,6$. Since the total number of substitution edges in H is even, we have $\left| {\mathcal {F}}_2 \right| + \left| {\mathcal {F}}_5 \right| + \left| {\mathcal {F}}_7 \right|$ is even.

Now, depending on the subscript $i \in \{1,2,\dots , 7\}$, we give a partition to every element of ${\mathcal {F}}_i$. (We do not require that it is a bisection for some i.) In order to do that, we first give a partition to each $C \in {\mathcal {F}}$.

Let $C \in {\mathcal {F}}$. Recall that C is an even cycle, and hence we denote the sequence of the vertices of C by $x_{1}, x_{2}, \dots , x_{2t}$, where $2t = |C|$. By symmetry, we may assume that $x_{2t} x_1 \in F_2$ and $x_1 x_2 \in F_1$. If C is an O-cycle, we may further assume that an end vertex of $e_C$ together with $x_{2t}$ and $x_1$ forms a triangle of H. Then, along C, we define $(B_C, W_C)$ as follows: First, let $x_{1} \in B_C$. Let $j = 1,2, \dots ,2t-1$.

• Suppose $x_{j} \in B_C$.

  • If $x_j x_{j+1}$ is a substitution edge, then add $x_{j+1}$ to $B_C$.

  • If $x_j x_{j+1}$ is not a substitution edge, then add $x_{j+1}$ to $W_C$.

• Suppose $x_{j} \in W_C$.

  • If $x_j x_{j+1}$ is a substitution edge, then add $x_{j+1}$ to $W_C$.

  • If $x_j x_{j+1}$ is not a substitution edge, then add $x_{j+1}$ to $B_C$.

We observe the following.

• Suppose that C is an E-cycle, that is, $C \in {\mathcal {F}}_1$. Since C is an even cycle with an even number of substitution edges, we have $x_{2t} \in W_C$, (see the left of Fig. 12). Since no edge in $E(C) \cap F_2$ is monochromatic, we have $|B_C| = |W_C|$.

• Suppose that C is an O-cycle. Then, $x_{2t} \in B_C$. Since no edge in $E(C) \cap F_2$ is monochromatic except for $x_{2t}x_1$, we have $|B_C| = |W_C| + 2$, (see the right of Fig. 12).

Fig. 12

A partition $(B_C, W_C)$ on an E-cycle and that on an O-cycle, where double lined edges represent substitution edges

We next consider $P \in {\mathcal {F}}_i$ for $i =2,3$. Let C be an O-cycle contained in P, let $x_1, x_2, \dots , x_{2t}$ be the sequence of the vertices of C, and let $e_C = x_Cy_C \in E(P) \cap M$ such that $x_C, x_{2t}$ and $x_1$ form a triangle of H. Then we define $(B_P, W_P)$ as follows (see Fig. 13):

$$\begin{aligned} B_P= & {} {\left\{ \begin{array}{ll} B_C \cup \{y_C\} &{} \hbox { for}\ P \in {\mathcal {F}}_2, \\ B_C &{} \hbox { for}\ P \in {\mathcal {F}}_3, \\ \end{array}\right. } \\ \text { and } W_P= & {} {\left\{ \begin{array}{ll} W_C \cup \{x_C\} &{} \hbox { for}\ P \in {\mathcal {F}}_2, \\ W_C \cup \{x_C, y_C\} &{} \hbox { for}\ P \in {\mathcal {F}}_3. \\ \end{array}\right. } \end{aligned}$$

Recall that $|B_C| = |W_C| + 2$. Thus, $|B_P| = |W_P|+2$ for $P\in {\mathcal {F}}_2$, while $|B_P| = |W_P|$ for $P\in {\mathcal {F}}_3$.

Fig. 13

A partition $(B_P, W_P)$ on $P \in {\mathcal {F}}_i$ for $i=2,3$

Let $P \in {\mathcal {F}}_i$ for $i =4,5$, let $C, C'$ be two O-cycles contained in P, and let $e_C = x_Cx_{C'} \in E(P) \cap M$ such that $x_C$ together with two vertices in C (resp. $x_{C'}$ together with two vertices in $C'$) forms a triangle of H. Then we define $(B_P, W_P)$ as follows (see Figs. 14 and 15):

$$\begin{aligned} B_P= & {} {\left\{ \begin{array}{ll} B_C \cup W_{C'} \cup \{x_{C'}\} &{} \hbox { for}\ P \in {\mathcal {F}}_4, \\ B_C \cup B_{C'} &{} \hbox { for}\ P \in {\mathcal {F}}_5, \\ \end{array}\right. }\\ \text { and } W_P= & {} {\left\{ \begin{array}{ll} W_C \cup B_{C'} \cup \{x_C\} &{} \hbox { for}\ P \in {\mathcal {F}}_4, \\ W_C \cup W_{C'} \cup \{x_C, x_{C'}\} &{} \hbox { for}\ P \in {\mathcal {F}}_5. \\ \end{array}\right. } \end{aligned}$$

Since $|B_C| = |W_C| + 2$ and $|B_{C'}| = |W_{C'}| + 2$, we see $|B_P| = |W_P|$ for $P\in {\mathcal {F}}_4$, while $|B_P| = |W_P|+2$ for $P\in {\mathcal {F}}_5$.

Fig. 14

A partition $(B_P, W_P)$ on $P \in {\mathcal {F}}_4$

Fig. 15

A partition $(B_P, W_P)$ on $P \in {\mathcal {F}}_5$

For each $P \in {\mathcal {F}}_{i}$ for $i = 6,7$, letting $V(P) = \{x,x'\}$, we define

$$\begin{aligned} B_P= & {} {\left\{ \begin{array}{ll} \{x\} &{} \hbox { for}\ P \in {\mathcal {F}}_6, \\ \{x, x'\} &{} \hbox { for}\ P \in {\mathcal {F}}_7, \\ \end{array}\right. } \text { and } W_P \ = \ {\left\{ \begin{array}{ll} \{x'\} &{} \hbox { for}\ P \in {\mathcal {F}}_6, \\ \emptyset &{} \hbox { for}\ P \in {\mathcal {F}}_7. \end{array}\right. } \end{aligned}$$

Note that $|B_P| = |W_P|$ for $P\in {\mathcal {F}}_6$, and $|B_P| = |W_P|+2$ for $P\in {\mathcal {F}}_7$.

Now, from $(B_P, W_P)$ for each $P \in \bigcup _{i=1}^7 {\mathcal {F}}_i$, we construct a bisection $(B_H, W_H)$ required in Claim 1. As explained before, $|{\mathcal {F}}_{2} |$ + $|{\mathcal {F}}_{5} |$ + $|{\mathcal {F}}_{7} |$ is even. So we can divide ${\mathcal {F}}_{2} \cup {\mathcal {F}}_{5} \cup {\mathcal {F}}_{7}$ into ${\mathcal {F}}^{+}$ and ${\mathcal {F}}^{-}$ such that ${\mathcal {F}}^{+} \cup {\mathcal {F}}^{-} = {\mathcal {F}}_{2} \cup {\mathcal {F}}_{5} \cup {\mathcal {F}}_{7}$, ${\mathcal {F}}^{+} \cap {\mathcal {F}}^{+}=\emptyset$ and $|{\mathcal {F}}^{+} |$ = $|{\mathcal {F}}^{-} |$. Now we define a partition $(B_H,W_H)$ of V(H) as follows:

$$\begin{aligned} B_H= & {} \bigcup _{P \in {\mathcal {F}}_{1} \cup {\mathcal {F}}_{3} \cup {\mathcal {F}}_{4} \cup {\mathcal {F}}_{6}}{B_P} \cup \bigcup _{P \in {\mathcal {F}}^{+}}{B_P} \cup \bigcup _{P \in {\mathcal {F}}^{-}}{W_P},\\ \text { and } W_H= & {} \bigcup _{P \in {\mathcal {F}}_{1} \cup {\mathcal {F}}_{3} \cup {\mathcal {F}}_{4} \cup {\mathcal {F}}_{6}}{W_P} \cup \bigcup _{P \in {\mathcal {F}}^{+}}{W_P} \cup \bigcup _{P \in {\mathcal {F}}^{-}}{B_P}. \end{aligned}$$

Recall that $|B_P| = |W_P|$ for $P \in {\mathcal {F}}_{1} \cup {\mathcal {F}}_{3} \cup {\mathcal {F}}_{4} \cup {\mathcal {F}}_{6}$, and $|B_P |= |W_P |+2$ for $P \in {\mathcal {F}}^{+} \cup {\mathcal {F}}^{-}$. Since $|{\mathcal {F}}^{+} |= |{\mathcal {F}}^{-} |$, we obtain that $|B_H| = |W_H|$, that is, $(B_H, W_H)$ is a bisection of H.

Let u, v, w be the three vertices of a triangle of H. Note that exactly one edge of the triangle is contained in $F_2$, say uv. Let $P \in \bigcup _{i=1}^5 {\mathcal {F}}_i$ such that $uv \in E(P)$. By the construction of $(B_P, W_P)$, the vertices u and v belong to distinct part of $(B_P, W_P)$, unless $\{u,v\} = \{x_{2t}, x_1\} \subseteq B_P$ for some O-cycle C contained in P with vertices $x_1, x_2, \dots ,x_{2t}$ along C, where $2t = |C|$. The exceptional case occurs only when $i = 2,3,4,5$, and in each of the case, we see $w = x_C \in W_P$. This implies that Condition (B1$'$) holds.

Note that each edge e in $E(H')$ is contained in some $P \in \bigcup _{i=1}^7 {\mathcal {F}}_{i}$. Thus, it follows from the construction of the partition $(B_P, W_P)$ that e is a monochromatic edge if and only if e is a substitution edge. Since each edge in $E(H) \setminus E(H')$ are contained in a triangle, Conditions (B2$'$) and (B3$'$) are satisfied.

This completes the proof of Claim 1. $\square$

By Claim 1, we obtain a bisection $(B_H, W_H)$ satisfying Conditions (B1$'$)–(B3$'$). By Conditions (B2$'$) and (B3$'$), an edge e in $H'$ is a monochromatic edge if and only if e is a substitution edge.

We next take all substitution edges back to strings of diamonds. Let $e=xy$ be a substitution edge in H, and let $A_e$ be a diamond of G such that e is replaced with $A_e$ when we obtain G from H. Let $a_{e}, b_{e}, c_{e}$ and $d_{e}$ be the vertices of $A_e$ such that $a_{e} d_{e} \notin E(G)$, and $x a_{e}, y d_{e} \in E(G)$.

By Condition (B3$'$), the end vertices x and y of e belong to the same part of $(B_H, W_H)$. If $x \in B_H$ (and hence $y \in B_H$), then we add $\{b_e,c_e\}$ to $B_H$ and $\{a_e,d_e\}$ to $W_H$. If $x,y \in W_H$, then we add $\{a_e,d_e\}$ to $B_H$ and $\{b_e,c_e\}$ to $W_H$. Note that neither the edge $xa_e$ nor the edge $y d_e$ are monochromatic.

Let B and W be the resultant set by the addition above for each substitution edge of H independently. Since $|B_H| = |W_H|$, we have $|B| = |W|$, that is, (B, W) is a bisection of G. By Condition (B1$'$) for $(B_H, W_H)$ and the definition of B  and  W, Condition (B1) is satisfied. Condition (B2$'$) for $(B_H, W_H)$ and the definition of B  and  W implies Condition (B2). The bisection (B, W) satisfies Condition (B3) by the definition of $B\, \textrm{and}\, W$. Condition (B4) trivially holds since G has no multiple edges. Thus, (B, W) is a desired bisection of G, contradicting the fact that G is minimum counterexample. This completes the proof of Theorem 3.2. $\square$