Graphs with a 3-Cycle-2-Cover

Abstract

If a graph $G$ has three even subgraphs $C_1$, $C_2$ and $C_3$ such that every edge of $G$ lies in exactly two members of $\{C_1, C_2, C_3\}$, then we say that $G$ has a 3-cycle-2-cover. Let $S_3$ denote the family of graphs that admit a 3-cycle-2-cover, and let $\mathcal {S}(h,k) = \{G:$ $G$ is at most $h$ edges short of being $k$-edge-connected$\}$. Catlin (J Gr Theory 13:465–483, 1989) introduced a reduction method such that a graph $G \in S_3$ if its reduction is in $S_3$; and proved that a graph in the graph family $\mathcal {S}(5,4)$ is either in $S_3$ or its reduction is in a forbidden collection consisting of only one graph. In this paper, we introduce a weak reduction for $S_3$ such that a graph $G \in S_3$ if its weak reduction is in $S_3$, and identify several graph families, including $\mathcal {S}(h,4)$ for an integer $h \ge 0$, with the property that any graph in these families is either in $S_3$, or its weak reduction falls into a finite collection of forbidden graphs.

1 Introduction

We study finite and loopless graphs with undefined terms and notations following Bondy and Murty [1]. For graphs $G$ and $H$, $H \subseteq G$ means that $H$ is a subgraph of $G$. If $X$ is an edge subset not in $G$ but every edge in $X$ has its end vertices in $G$, then $G + X$ is the graph with vertex set $V(G)$ and edge set $E(G) \bigcup X$. For a graph $G$, let $\kappa '(G)$ denote the edge-connectivity of $G$. A circuit is defined to be a nontrivial 2-regular connected graph, and a cycle to be an edge-disjoint union of circuits. A circuit of length $n$ will be denoted as $C^n$. Often a cycle is also called an even graph. A $3$-cycle-$2$-cover of $G$ is a collection of $3$ cycles of $G$ such that each edge of $G$ is in exactly two cycles of the collection.

The study of graphs with a $3$-cycle-$2$-cover is motivated by the theory of nowhere zero flows, initiated by Tulle [23] more than half a century ago. Let $D=D(G)$ be an orientation of a graph $G$. For a vertex $v \in V(D)$, let $E_D^+(v)$ ($E_D^-(v)$, respectively) denote the set of all edges oriented outgoing from $v$ (oriented incoming into $v$, respectively). Let $k > 1$ be an integer. A function $f$ from $E(D)$ to the set of integers is a nowhere zero $k$-flow if for any $e \in E(D)$, $f(e) \ne 0$ and $|f(e)| < k$ and for any $v \in V(D)$, $\sum _{e \in E_D^+(v)} f(e) = \sum _{e \in E_D^-(v)} f(e)$. It is well known (for example, see [5, 15, 22]) that a connected graph $G$ admitting a nowhere zero $4$-flow if and only if $G$ has a $3$-cycle-$2$-cover.

For a graph $G$, let $O(G)$ be the set of odd-degree vertices of $G$. Thus $G$ is a cycle if and only if $O(G) = \emptyset$. A graph $G$ is collapsible ([4], see also Proposition 1 of [17]) if for every subset $R \subseteq V(G)$ with $|R|$ even, $G$ has a subgraph $\Gamma _R$ such that $O(\Gamma _R) = R$ and $G - E(\Gamma _R)$ is connected. Following Catlin [5], we use $\mathcal {CL}$ to denote the family of collapsible graphs. An edge subset $X\subseteq E(G)$ is an $O(G)$-join if $O(G[X])=O(G)$. We have the following observations.

Observation 1.1

Let $G$ be a graph.

• (i) An edge subset $X\subseteq E(G)$ is an $O(G)$-join of $G$ if and only if $G-X$ is a cycle.

• (ii) If $E(G)=E_{1} \bigcup E_{2}\bigcup E_{3}$ is a disjoint union of 3 $O(G)$-joins, then $G$ has $3$ cycles $C_i = G-E_{i}$, $i=1,2,3$, such that every edge $e\in E(G)$ is in exactly two members of the set (possibly a multiset) $\{C_1, C_2, C_3\}$. (In this case, $\{C_1, C_2, C_3\}$ is a $3$-cycle-$2$-cover of $G$).

Following Catlin [5], we define $S_{3}$ to be the family of connected graphs admitting a $3$-cycle-$2$-cover. A graph $G$ in $S_{3}$ will be called an $S_{3}$-graph. As mentioned above, $S_{3}$ is the family of connected graphs that admit nowhere zero $4$-flows.

Jaeger [14] proved that every $4$-edge-connected graph is in $S_{3}$. It is known (see [5, 15, 22]) that $3$-edge-connectedness does not warrant a membership in $S_{3}$, as evidenced by the Petersen graph. Hence, characterizing $S_{3}$-graphs among $3$-edge connected graphs has been a problem for investigation. Such problem is not just interesting by itself, it is also closely related to the study on Chinese Postman problem and Traveling Salesman problem [2].

Catlin in [5] defined a graph reduction and identified a family $\mathcal {F}$ of $3$-edge-connected graphs that are closed to be $4$-edge-connected, with the property that a graph $G \in \mathcal {F}$ is either in $S_3$ or its reduction is in $\{P(10)\}$, where $P(10)$ is the Petersen graph.

Graph contraction is needed to describe Catlin’s reduction. For $X\subseteq E(G)$, the contraction $G/X$ is the graph obtained from $G$ by identifying the two ends of each edge in $X$ and then deleting the resulting loops. We define $G/\emptyset =G$. If $H \subseteq G$, then we write $G/H$ for $G/E(H)$. If $H$ is a connected subgraph of $G$, and if $v_H$ is the vertex in $G/H$ onto which $H$ is contracted, then $H$ is the preimage of $v_H$, and is denoted by $PI_G({v_H})$. Given a family $\mathcal {F}$ of connected graphs, for any graph $G$, an $\mathcal {F}$-reduction of $G$ is obtained from $G$ by successively contracting nontrivial subgraphs in $\mathcal {F}$ until none left.

Catlin in [4] showed that every graph $G$ has a unique collection of maximal collapsible subgraphs $H_1, H_2, \cdots , H_c$, and the ${\mathcal {CL}}$-reduction of $G$ is exactly $G'= G/(\cup _{i=1}^c E(H_i))$, which is unique. For a family $\mathcal {F}$ of graphs, Catlin in [7] defined

$$\begin{aligned} \mathcal {F}^{o}&= \{H|H \text{ is } \text{ connected, } \text{ and } \text{ for } \text{ graph } G \hbox { with } H \subseteq G, G/H \in \mathcal {F}\nonumber \\&\hbox { if and only if } G \in \mathcal {F}\}. \end{aligned}$$

(1.1)

Let $C^4$ denote a circuit of length $4$. For the family $S_3$, Catlin [5] showed ${\mathcal {CL}}\bigcup \{C^4\} \subseteq S_3^o$. In [5], Catlin defined, for integers $k, t > 0$,

$$\begin{aligned}&\mathcal {S}(h,k) = \{G: \text{ for } \text{ some } \text{ edge } \text{ set } X \cap E(G) = \emptyset \hbox { with } |X| \le h,\nonumber \\&\quad \hbox { and } \kappa '(G+X) \ge k \}. \end{aligned}$$

(1.2)

Theorem 1.2

(Catlin, Theorem 14 of [5]) Let $G$ be a graph in $\mathcal {S}(5,4)$. Then exactly one of the following holds:

• (i) $G\in S_{3}$.

• (ii) $G$ has at least one cut-edge.

• (iii) The ${\mathcal {CL}} \bigcup \{C^4\}$-reduction of $G$ is the Petersen graph.

Theorem 1.2 indicates that within certain graph families, one can characterize $S_{3}$-graphs in term of excluding a finite list of reductions. The purpose of this paper is to continue such investigation by studying more general families of graphs and to give a characterization of $S_{3}$-graphs within these families by excluding a finite list of certain reductions. To this aim, we define, for integers $h,k>0$,

$$\begin{aligned} N_h(k)=\{G: G \text{ is } \text{ simple }, |V(G)|\le k, \kappa '(G)\ge h, \hbox { and } G \notin S_{3} \}. \end{aligned}$$

In Theorem 3.10 of [9], it is shown that under certain general and necessary condition of $\mathcal {F}$, the $\mathcal {F}^o$-reduction is unique. In particular, the $S_3^o$-reduction of any graph $G$ is uniquely determined by $G$. We in the next section will define a weak reduction for the family $S_3$ (called weak $S_3$-reduction) in which we might not have the uniqueness.

Suppose that $a, b$ are real numbers with $0<a<1$, and $f_{a,b}(n)=an+b$ is a function of $n$. Let $C(h,a,b)$ denote the family of simple graphs $G$ of order $n$ with $\kappa '(G)\ge h$ such that for any edge cut $X$ of $G$ with $|X|\le 3$, each component of $G-X$ has at least $f_{a,b}(n)$ vertices.

If a graph $G$ has a spanning eulerian subgraph, then $G$ is supereulerian. It is well known that all supereulerian graphs are in $S_3$ (see, for example, Section 7 of [6]). The prior results of graph families $C(h,a,b)$ are summarized in the theorem below.

Theorem 1.3

Let $G \in C(h,a,b)$ be a graph. Then each of the following holds.

• (i) (Catlin and Li [11]) If $h = 2, a = \frac{1}{5}$ and $b =0$, then $G$ is supereulerian or the reduction of $G$ is in $\{K_{2,3}, K_{2,5}\}$. Hence in any case, $G \in S_3$.

• (ii) (Broersma and Xiong [3]) If $h = 2, a = \frac{1}{5}$ and $b = -\frac{2}{5}$, then $G$ is supereulerian or the reduction of $G$ is in a family of 3 exceptional cases, all of which are in $S_3$.

• (iii) (Li et al, [18]) If $h = 2, a = \frac{1}{6}$ and $b = -\frac{2}{5}$, then $G$ is supereulerian or the reduction of $G$ is in a finite family of exceptional cases. Thus any such $G$ is in $S_3$ if and only if the $\mathcal {CL}$-reduction of $G$ is not in a finite forbidden family of graphs.

• (iv) (Lai and Liang [16]) If $h = 2, a = \frac{1}{6}$ and $b$ is any fixed number, then $G$ is supereulerian or the reduction of $G$ is in a finite family of exceptional cases. Thus any such $G$ is in $S_3$ if and only if the $\mathcal {CL}$-reduction of $G$ is not in a finite forbidden family of graphs.

• (v) (Li et al [19]) If $h = 2, a = \frac{1}{7}$ and $b=0$, then $G$ is supereulerian or the reduction of $G$ is in a finite family of exceptional cases. Thus any such $G$ is in $S_3$ if and only if the $\mathcal {CL}$-reduction of $G$ is not in a finite forbidden family of graphs.

• (vi) (Niu and Xiong [21]) If $h = 3, a = \frac{1}{10}$ and $b$ is any fixed number, then $G$ is supereulerian or the reduction of $G$ is in a finite family of exceptional cases. Thus any such $G$ is in $S_3$ if and only if the $\mathcal {CL}$-reduction of $G$ is not in a finite forbidden family of graphs.

Theorems 1.2 and 1.3 motivate our research. The main results of this paper are the following.

Theorem 1.4

Let $G$ be a graph of order $n$. For any real numbers $a$ and $b$ with $0 < a < 1$, if $G \in C(2,a,b)$, then one of the following holds.

• (i) $G \in S_{3}$.

• (ii) Every weak $S_{3}$-reduction of $G$ is in $N_2(\lceil \frac{3}{a}\rceil )$.

For a graph $G$, let $t_3(G)$ be the number of $3$-edge-cuts of $G$. For a given integer $k$, define

$$\begin{aligned} \mathcal{W}(k)=\{ G\ | \ G \text{ is } \text{ simple } \text{ and } t_3(G) \le k\}. \end{aligned}$$

Theorem 1.5

Let $G$ be a graph of order $n$ with $\kappa '(G)\ge 3$. For a given integer $k \ge 0$, if $G\in \mathcal{W}(k)$, then one of the following holds.

• (i) $G\in S_{3}$.

• (ii) $k \ge 10$ and every weak $S_{3}$-reduction of $G$ is in $N_3(2k-10)$.

Theorem 1.6

Let $G$ be a graph of order $n$. For an integer $h \ge 0$, if $G\in \mathcal {S}(h,4)$ satisfies $\kappa '(G)\ge 3$, then one of the following holds.

• (i) $G\in S_{3}$.

• (ii) $h \ge 5$, and every weak $S_{3}$-reduction of $G$ is in $N_3(4h-10)$.

It is well known that the Petersen graph is the only 3-edge-connected graph with at most 10 vertices that is not in $S_3$. Hence when $h=5$, Theorem 1.6 implies that a graph $G \in \mathcal {S}(5,4)$ is not in $S_3$ if and only if the only weak $S_{3}$-reduction of $G$ is the Petersen graph. This fact relates our result to Catlin’s Theorem 14 of [5]. Furthermore, for given $a$, $b$, $k$ and $h$, each graph in $N_2(\lceil \frac{3}{a}\rceil )\cup N_3(2k-10)\cup N_3(4h-10)$ has order independent on $n$. Thus, the number of graphs in $N_2(\lceil \frac{3}{a}\rceil )\cup N_3(2k-10)\cup N_3(4h-10)$ is fixed and finite. From a computational point of view, for given $a$, $b$, $k$ and $h$, each of these families: $N_2(\lceil \frac{3}{a}\rceil )$ or $N_3(2k-10)$ or $N_3(4h-10)$, can be determined in a constant time. Like the characterization of planar graphs, people view that $K_5$ and $K_{3,3}$ are the only two nonplanar graphs. By Theorems 1.4, 1.5 and 1.6, in some sense, we can see that only a finite number of graphs in $C(2,a,b)$ or 3-edge-connected graphs in $\mathcal{W}(k)\cup \mathcal {S}(h,4)$ are not in $S_{3}$.

In Sect. 2, weak $S_{3}$-reduction of graphs will be introduced and certain reduction results will be reviewed and developed. The proofs of the main theorems are given in the last section.

2 Reductions

We will introduce weak $S_{3}$-reduction of graphs in this section. Let $G$ be a graph and $i \ge 0$ be an integer. Define

$$\begin{aligned} V_{i}(G)=\{v\in V(G)|d_{G}(v)=i\}; \quad \text{ and } \quad d_{i}(G)=|V_{i}(G)|. \end{aligned}$$

For a vertex $v \in V(G)$, $N_G(v)$, the neighborhood of $v$, is the set of vertices adjacent to $v$ in $G$. For a vertex $u \in V(G)$ with $N_G(u) = \{v_{1},v_{2},v_{3},v_{4}\}$, let $\pi =\langle \{v_{i_1},v_{i_2}\},\{v_{i_3},v_{i_4}\}\rangle$ be a $2$-partition of $N_G(u)$ into a pair of 2-subsets. Define $G_{\pi }$ to be the graph obtained from $G-u$ by adding new edges $v_{i_1}v_{i_2}, v_{i_3}v_{i_4}$. We say that $G_{\pi }$ is obtained from $G$ by dissolving $u$ (via a $2$-partition $\pi$).

Theorem 2.1

(Fleischer [12], Mader [20]) If $u \in V_{4}(G)$ with $|N_G(u)| = 4$, then for some $2$-partition $\pi$ of $N_G(u)$, $\kappa '(G_{\pi })=\kappa '(G)$.

Theorem 2.2

(Catlin) Let $G$ be a graph, $H$ be a collapsible subgraph of $G$, $G_{\pi }$ be the graph obtained from $G$ by dissolving a vertex $u\in V_{4}(G)$, and $G'$ be the $\mathcal {CL}$-reduction of $G$. Then each of the following holds.

• (i) (Corollary 13A of [5]) $\mathcal {CL}\cup \{C^4\} \subset S_3^o$. In particular, $G' \in S_{3}$ if and only if $G\in S_{3}$.

• (ii) (Lemma 3 of [5]) If $G_{\pi }\in S_{3}$, then $G\in S_{3}$.

• (iii) (Theorem 8 of [4]) $G'$ is simple.

For a graph $G$, let $F(G)$ be the minimum number of additional edges that must be added to $G$ to result in a graph with 2-edge-disjoint spanning trees. The following has been proved.

Theorem 2.3

Let $G$ be a connected graph. Each of the following holds.

• (i) (Catlin, Theorem 7 of [4]) If $F(G)\le 1$, then either $G$ is collapsible or the reduction of $G$ is $K_2$.

• (ii) (Catlin et al, Theorem 1.3 of [8]) If $F(G)\le 2$, then either $G$ is collapsible, or the reduction of $G$ is a $K_2$ or a $K_{2,t}$ for some integer $t\ge 1$.

It follows from Theorems 2.2 and 2.3 that

$$\begin{aligned} \text{ if } \kappa '(G)\ge 2 \quad \hbox { and }\quad F(G)\le 2, \hbox { then } G \in S_{3}. \end{aligned}$$

(2.1)

Let $G'$ be the ${\mathcal {CL}}$-reduction of $G$. By Lemma 2.3 of [8], we have

$$\begin{aligned} F(G')=2|V(G')|-|E(G')|-2. \end{aligned}$$

(2.2)

As $|V(G')|= \sum _{i \ge 1} d_{i}(G')$ and $2|E(G')|=\sum _{i \ge 1} id_{i}(G')$, it follows from (2.2) that

$$\begin{aligned} 2F(G')=4\sum _{i\ge 1}d_{i}(G')-\sum _{i\ge 1}id_{i}(G')-4 =\sum _{i\ge 1}(4-i)d_{i}(G')-4, \end{aligned}$$

and so

$$\begin{aligned} 3d_{1}(G')+2d_{2}(G')+d_{3}(G')=2F(G')+4+\sum _{i \ge 5}(i-4)d_{i}(G'). \end{aligned}$$

(2.3)

Let $G$ be a graph and $G'$ be the ${\mathcal {CL}}$-reduction of $G$. A weak $S_3$-reduction of $G$ is obtained from $G'$ by repeatedly dissolving vertices of degree 4 in $G'$ while preserving the edge-connectivity of $G'$, until no vertices of degree 4 are left. Parts (i) and (ii) of the following lemma are immediate consequences of the definition of weak $S_3$-reduction and Theorem 2.2. Part (iii) is a consequence of (2.3) and Part (i).

Lemma 2.4

Let $G'$ be the $\mathcal {CL}$-reduction of $G$ and $G''$ be a weak $S_3$-reduction of $G$.

• (i) $V_4(G'') = \emptyset$, and for any $i \ne 4$, $d_i(G'') = d_i(G')$.

• (ii) If $G'' \in S_3$, then $G \in S_3$.

• (iii) $3d_{1}(G'')+2d_{2}(G'')+d_{3}(G'')=2F(G')+4+\sum _{i \ge 5}(i-4)d_{i}(G'')$. In particular, if $\kappa '(G) \ge 3$, then $d_3(G'') = 2F(G') + 4+\sum _{i \ge 5}(i-4)d_{i}(G'')$.

To prove our main results, we need to show that

$$\begin{aligned} \text{ a } \text{ graph } G \hbox { is in } S_3 \hbox { if and only if } G \hbox { has one weak } S_3\hbox {-reduction in } S_3. \end{aligned}$$

(2.4)

Theorem 2.2 indicates that if a weak $S_3$-reduction of $G$ is in $S_3$, then $G \in S_3$. To show the necessity of (2.4), we will prove the following lemma to justify (2.4).

Lemma 2.5

Let $G$ be a connected graph. If $G \in S_3$, then $G$ has one weak $S_3$-reduction in $S_3$.

Proof

Let $G \in S_3$, and let $G'$ be the $\mathcal {CL}$-reduction of $G$. By Theorem 2.2, $G' \in S_3$. We shall show that a weak reduction $G''$ of $G$ is in $S_3$. If $V_4(G')=\emptyset$, then $G''=G'$ is the weak $S_3$-reduction of $G$. As $G' \in S_3$, we are done. Hence we argue by induction on $|V_4(G')|$ and assume that $V_4(G') \ne \emptyset$.

Pick a vertex $u \in V_4(G')$. By Theorem 2.2, $G'$ is simple and so we may assume that $N_{G'}(u)=\{v_1,v_2,v_3,v_4\}$ and $E_{G'}(u)=\{uv_1,uv_2,uv_3,uv_4\}$. To complete inductive argument, we shall find a 2-partition $\pi$ of $N_{G'}(u)$ such that $G'_{\pi } \in S_3$. Note that by the definition of $G'_{\pi }$, we can view $V(G') - \{u\} = V(G'_{\pi })$. As $u \in V_4(G')$, $O(G') = O(G'_{\pi })$.

Since $G'\in S_3$, there exist edge-disjoint $O(G')$-joins $E_1', E_2', E_3' \subseteq E(G')$ such that $E_1' \bigcup E_2' \bigcup E_3'=E(G')$. For $i=1,2,3$, since $u \notin O(G')$ and since $E_i'$ is an $O(G')$-join, $|E_{G'}(u) \bigcap E_i'| \equiv 0$ (mod 2). Since $\{E_1', E_2', E_3'\}$ is a partition of $E(G')$, we may assume that either $E_{G'}(u) \subseteq E_1'$ and $|E_{G'}(u) \bigcap E_i'| = 0$ for $i \in \{2,3\}$, or $|E_{G'}(u) \bigcap E_1'| = |E_{G'}(u) \bigcap E_2'| = 2$ and $|E_{G'}(u) \bigcap E_3'| = 0$.

Case 1. $E_{G'}(u) \subseteq E_1'$ and $|E_{G'}(u) \bigcap E_i'| = 0$ for $i \in \{2,3\}$.

Define $\pi =\langle \{v_{1},v_{2}\},\{v_{3},v_{4}\}\rangle$, and let $E_1''= (E_1' - E_{G'}(u)) \bigcup \{v_1v_2,v_3v_4\}$, $E''_2=E'_2$ and $E''_3=E'_3$. As $O(G') = O(G'_{\pi })$, each $E_i''$ is an $O(G'_{\pi })$-join. Since $E_1', E_2', E_3'$ are edge-disjoint in $E(G')$ with $E_1' \bigcup E_2' \bigcup E_3'=E(G')$, we conclude that $E_1'', E_2'', E_3''$ are edge-disjoint in $E(G'_{\pi })$ with $E_1'' \bigcup E_2'' \bigcup E_3''=E(G'_{\pi })$. By definition, $G'_{\pi } \in S_3$.

Case 2. $|E_{G'}(u) \bigcap E_1'| = |E_{G'}(u) \bigcap E_2'| = 2$ and $|E_{G'}(u) \bigcap E_3'| = 0$.

Without loss of generality, we assume that $uv_1,uv_2\in E'_1$ and $uv_3,uv_4\in E'_2$. Define $\pi =\langle \{v_{1},v_{2}\},\{v_{3},v_{4}\}\rangle$, and let $E_1''= (E_1' - E_{G'}(u)) \bigcup \{v_1v_2\}$, $E_2''= (E_2' - E_{G'}(u)) \bigcup \{v_3v_4\}$ and $E''_3=E'_3$. As $O(G') = O(G'_{\pi })$, each $E_i''$ is an $O(G'_{\pi })$-join. Since $E_1', E_2', E_3'$ are edge-disjoint in $E(G')$ with $E_1' \bigcup E_2' \bigcup E_3'=E(G')$, we conclude that $E_1'', E_2'', E_3''$ are edge-disjoint in $E(G'_{\pi })$ with $E_1'' \bigcup E_2'' \bigcup E_3''=E(G'_{\pi })$. By definition, $G'_{\pi } \in S_3$.

As in either case, we can always find a 2-partition $\pi$ of $N_{G'}(u)$ such that $G'_{\pi } \in S_3$, the lemma is proved by induction. $\square$

3 Proof of The Main Results

We shall prove the main results in this section. Throughout this section, $a, b$ denote two real numbers with $0 < a < 1$, and $h, k > 0$ denote two integers. Let $G$ be a graph in $C(2,a,b) \cup S(h,4) \cup \{G:\kappa '(G)\ge 2, t_3(G) \le k\}$. Assume that $G$ is not in $\mathcal{S}_3$, by (2.1), we have $F(G')\ge 3$. Let $G''$ be a weak $S_3$-reduction of $G$. We shall show that $|V(G'')|$ must be bounded by the quantities given in Theorems 1.4, 1.5 and 1.6, respectively. To simplify notations, for each $i$, let $d_i = d_i(G'')$.

Proof of Theorem 1.4

Assume first that $G \in C(2,a,b)$. By Lemma 2.4 (i) and (iii) and by $\kappa '(G) \ge 2$, we have

$$\begin{aligned} 2(d_{2}+d_{3}) \ge 2d_{2}+d_{3}=2F(G')+4+ \sum _{i \ge 4}(i-4)d_{i}. \end{aligned}$$

(3.1)

By (2.1) and (3.1), we have

$$\begin{aligned} 2(d_{2}+d_{3}) \ge 10+\sum _{i \ge 4}(i-4)d_{i}\ge 10+\sum _{i \ge 5}d_{i}. \end{aligned}$$

(3.2)

By the definition of $C(2,a,b)$, then the edges incident to a vertex of degree two (or three) in $G'$ correspond to a $2$-edge-cut (or $3$-edge-cut) in $G$. We have $(d_2+d_3) (an+b)\le n$, and so $\displaystyle d_2+d_3\le \frac{n}{an+b}\le \left\lceil \frac{1}{a}\right\rceil$ (if $b<0$, $n>-\frac{b}{a}(1+\frac{1}{a})$). It follows by (3.2) that

$$\begin{aligned} |V(G'')|=(d_2+d_3)+ \sum \limits _{i\ge 5} d_i\le 3 (d_2+d_3)\le \left\lceil \frac{3}{a}\right\rceil , \end{aligned}$$

which implies Theorem 1.4. $\square$

Proof of Theorem 1.5

Next we assume that $\kappa '(G) \ge 3$ and $t_3(G) \le k$. By the definition of contraction, every $3$-edge-cut of $G'$ is a $3$-edge-cut of $G$, and so $k\ge t_3(G)\ge t_3(G')\ge d_3.$ By Lemma 2.4 (i) and (iii) and $\kappa '(G)\ge 3$, we have

$$\begin{aligned} k\ge d_{3}=2F(G')+4+\sum _{i\ge 5}(i-4)d_{i}. \end{aligned}$$

By (2.1) and $\kappa '(G)\ge 3$, we have $F(G')\ge 3,$ and

$$\begin{aligned} k-10\ge d_3-10\ge \sum \limits _{i\ge 5} (i-4)d_i. \end{aligned}$$

It follows that

$$\begin{aligned} |V(G'')|=d_3+\sum \limits _{i\ge 5} d_i \le d_3+(d_3-10)\le 2k-10, \end{aligned}$$

which implies Theorem 1.5. $\square$

Proof of Theorem 1.6

Assume that $G\in S(h,4)$ with $\kappa '(G)\ge 3$. By the definition of $S(h,4)$, for any $G\in S(h,4)$, there exists an edge subset $X$ not in $G$ such that $\kappa '(G+X)\ge 4$ with $|X|\le h.$ Since $\delta (G+X)\ge \kappa '(G+X)\ge 4,$ we have $d_3\le 2h$. By Lemma 2.4 (i) and (iii), we have

$$\begin{aligned} d_{3}=2F(G')+4+\sum _{i\ge 5}(i-4)d_{i}. \end{aligned}$$

(3.3)

By (2.1), $F(G')\ge 3.$ This, together with (3.3), implies

$$\begin{aligned} d_{3}\ge 10+\sum _{i\ge 5}(i-4)d_{i}\ge 10+\sum _{i\ge 5}d_{i}. \end{aligned}$$

(3.4)

By (3.4),

$$\begin{aligned} |V(G'')|=d_3+\sum _{i\ge 5}d_{i}\le 2h+2h-10=4h-10, \end{aligned}$$

which implies Theorem 1.6. $\square$