On s-hamiltonicity of net-free line graphs

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Abstract

For integers s1,s2,s3>0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2 and s3, respectively. We prove the following results.

(i) Let N1={Ns1,s2,s3:s1>0,s1s2s30 and s1+s2+s36}. Then for any NN1, every N-free line graph L(G) with |V(L(G))|s+3 is s-hamiltonian if and only if κ(L(G))s+2.

(ii) Let N2={Ns1,s2,s3:s1>0,s1s2s30 and s1+s2+s34}. Then for any NN2, every N-free line graph L(G) with |V(L(G))|s+3 is s-Hamilton-connected if and only if κ(L(G))s+3.

Introduction

We consider finite graphs without loops but permitting multiple edges, and follow [1] for undefined terms and notations. In particular, for a graph G, κ(G), κ(G), δ(G) and Δ(G) denote the connectivity, edge-connectivity, the minimum degree and the maximum degree of G, respectively. We use c(G) and g(G) to denote the circumference and the girth of G, which are the length of a longest cycle in G and the length of a shortest cycle of G, respectively. A graph is trivial if it has no edges. We write HG to mean that H is a subgraph of G. If XE(G), then G[X] is the subgraph of G induced by X. If H and K are subgraphs of a graph G, then we define HK=G[E(H)E(K)]. Throughout this paper, we use Pk to denote a path of order k. For integers s1,s2,s30, let Ns1,s2,s3 denote the graph formed by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2, and s3, respectively. A graph G is {H1,H2,,Hs}-free if G contains no induced subgraph isomorphic to any copy of Hi for any i. If s=1, then an {H1}-free graph is simply called an H1-free graph. A claw-free graph is just a K1,3-free graph. As in [1], a graph is hamiltonian if it has a spanning cycle and is Hamilton-connected if every pair of distinct vertices is joined by a spanning path.

The line graph of a graph G, denoted by L(G), is a simple graph with vertex set E(G), where two vertices in L(G) are adjacent if and only if the corresponding edges in G are adjacent. A few most fascinating problems in this area are presented below. By an ingenious argument of Z. Ryjác̆ek [32], Conjecture 1.1(i) is equivalent to a seeming stronger conjecture of Conjecture 1.1(ii). In [33], it is shown that all conjectures stated in Conjecture 1.1 are equivalent to each other.

Conjecture 1.1

(i) (Thomassen [35]) Every 4-connected line graph is hamiltonian.

(ii) (Matthews and Sumner [30]) Every 4-connected claw-free graph is hamiltonian.

(iii) (Kučel and Xiong [19]) Every 4-connected line graph is Hamilton-connected.

(iv) (Ryjáček and Vrána [33]) Every 4-connected claw-free graph is Hamilton-connected.

Towards Conjecture 1.1, Zhan gave a first result in this direction, and the best known result is given by Kaiser and Vrána, as shown below.

Theorem 1.2

Let G be a graph.

(i) (Zhan, Theorem 3 in [37]) If κ(L(G))7, then L(G) is Hamilton-connected.

(ii) (Kaiser and Vrána [18]) Every 5-connected claw-free graph with minimum degree at least 6 is hamiltonian.

(iii) (Kaiser, Ryjáček and Vrána [17]) Every 5-connected claw-free graph with minimum degree at least 6 is 1-Hamilton-connected.

There have been many researches on hamiltonian properties in 3-connected claw-free graphs forbidding a Nk,0,0, as seen in the surveys in [2], [11], [13], [14], among others. The following have been proved.

Theorem 1.3

Let Q be the graph obtained from the Petersen graph by adding one pendant edge to each vertex. Let G be a 3-connected simple claw-free graph.

(i) (Brousek, Ryjáček and Favaron, [4]) If G is N4,0,0-free, then G is hamiltonian.

(ii) [24] If G is N8,0,0-free, then G is hamiltonian. Moreover, the graph Q indicates the sharpness of this result.

(iii) (Fujisawa, [12], see also Ma et al. [29]) If G is N9,0,0-free graph, then G is hamiltonian unless G is the line graph of Q.

It is natural to seek necessary and sufficient conditions for hamiltonicity of line graphs. For an integer s0, a graph G of order ns+3 is s-hamiltonian (s-Hamilton-connected, respectively), if for any XV(G) with |X|s, GX is hamiltonian (GX is Hamilton-connected, respectively). It is well known that if a graph G is s-hamiltonian, then G is (s+2)-connected, and if G is s-Hamilton-connected, then G is (s+3)-connected. Broersma and Veldman in [3] initiated the problem of investigating graphs whose line graph is s-hamiltonian if and only if the connectivity of the line graph is at least s+2. They define, for an integer k0, a graph G to be k-triangular if every edge of G lies in at least k triangles of G. The following is obtained.

Theorem 1.4 Broersma and Veldman, [3]

Let ks0 be integers and let G be a k-triangular simple graph. Then L(G) is s-hamiltonian if and only L(G) is (s+2)-connected.

Broersma and Veldman in [3] proposed an open problem of determining the range of an integer s such that within triangular graphs, L(G) is s-hamiltonian if and only L(G) is (s+2)-connected. This problem was first settled by Chen et al. in [10].

Theorem 1.5

Each of the following holds.

(i) (Chen et al. [10]) Let k and s be positive integers such that 0smax{2k,6k16}, and let G be a k-triangular simple graph. Then L(G) is s-hamiltonian if and only L(G) is (s+2)-connected.

(ii) [21] Let G be a connected graph and let s5 be an integer. Then L(G) is s-hamiltonian if and only if L(G) is (s+2)-connected.

An hourglass is a graph isomorphic to K5E(C4), where C4 is a cycle of length 4 in K5. The following are proved recently.

Theorem 1.6

Each of the following holds.

(i) (Kaiser, Ryjáček and Vrána [17]) Every 4-connected claw-free hourglass-free graph is 1-Hamilton-connected.

(ii) [25] For an integer s2, the line graph L(G) of a claw-free graph G is s-hamiltonian if and only if L(G) is (s+2)-connected.

(iii) [25] The line graph L(G) of a claw-free graph G is 1-Hamilton-connected if and only if L(G) is 4-connected.

(iv) (Hu and Zhang [16]) Every 3-connected {K1,3,N1,2,3}-free graph is Hamiltonian-connected.

In view of Conjecture 1.1 and motivated by Theorem 1.2, Theorem 1.3, Theorem 1.5, Theorem 1.6, it is conjectured [21] that for any integer s2, L(G) is s-hamiltonian if and only if κ(L(G))s+2. The main goal of this research is to investigate if N8,0,0 in Theorem 1.3(ii) can be replaced by other Ns1,s2,s3 and if further evidences to support the conjecture in [21] can be found. The following results are obtained.

Theorem 1.7

Let s be an integer.

(i) Let N1={Ns1,s2,s3:s1>0,s1s2s30 and s1+s2+s36}. Then for any NN1, every N-free line graph L(G) with |V(L(G))|s+3 is s-hamiltonian if and only if κ(L(G))s+2 for s>0.

(ii) Let N2={Ns1,s2,s3:s1>0,s1s2s30 and s1+s2+s34}. Then for any NN1, every N-free line graph L(G) with |V(L(G))|s+3 is s-Hamilton-connected if and only if κ(L(G))s+3 for s0.

Theorem 1.7 extends Theorem 1.3(i) in the context of line graph and furthers the main results in [36]. Let O(G) be the set of odd degree vertices of a graph G. Following [1], a graph G is eulerian if G is connected with O(G)=. A graph G is supereulerian if G contains a spanning eulerian subgraph. To prove Theorem 1.7, we prove an auxiliary theorem (Theorem 3.2 in Section 4), which leads to the following extension of Theorem 4 in [24].

Theorem 1.8

Let G be a 2-edge-connected graph. Each of the following holds.

(i) Let Γ be a graph with κ(Γ)3 and eE(Γ). If G=Γe and c(G)8, then G is supereulerian.

(ii) If c(G)8 and G has at most two edge-cuts of size 2, then G is supereulerian.

Preliminaries and tools will be presented in the next section. In Sections 3 Auxiliary theorem and the proof of, 4 Proof of, we assume the validity of a auxiliary theorem (Theorem 3.2 in Section 4) to prove Theorem 1.8, Theorem 1.7, respectively. Theorem 3.2 will be proved in the last section.

Section snippets

Preliminaries

In [6] Catlin introduced collapsible graphs. It is shown in Proposition 1 of [22]) that a graph G is collapsible if for every subset RV(G) with |R|=0 (mod 2), G has a spanning connected subgraph Γ such that O(Γ)=R. See Catlin’s survey [7] and it supplements [8], [22] for further literature in this area. We use the notation that for a graph G and an integer i0, define Di(G)={vV(G):dG(v)=i}.

For a graph G and XE(G), the contraction GX is the graph formed from G by contracting edges in X with

Auxiliary theorem and the proof of Theorem 1.8

We first present an auxiliary theorem, stated as Theorem 3.2. For notational convenience, we define c(K1)=0. We will assume the validity of Theorem 3.2 to prove Theorem 1.8. The justification of Theorem 3.2 will be postponed to the last section. We start with a lemma.

Lemma 3.1

Let G be a connected graph with a 2-edge-cut X and let G1 and G2 be the two components of GX. If both GG1 and GG2 are supereulerian, then G is also supereulerian.

Proof

For i{1,2}, let vi denote the vertex in GGi onto which Gi is

Proof of Theorem 1.7

In this section, we assume the validity of Theorem 3.2 to prove Theorem 1.7. For an integer m>0, we use Zm to denote the cyclic group of order m. For integers s1s2s31, let Ys1,s2,s3 be the graph obtained from disjoint paths Ps1+2,Ps2+2 and Ps3+2 by identifying an end vertex of each of these three paths. (See Fig. 1 in [36] for an example.) By definition, Ns1,s2,s3=L(Ys1,s2,s3). Define Y1={Ys1,s2,s3:s1>0,s1s2s30,s1+s2+s36}.Y2={Ys1,s2,s3:s1>0,s1s2s30,s1+s2+s34}. By definition of line

Proofs of Lemma 4.4 and Theorem 3.2

The arguments in this section do not depend on any result in Sections 3 Auxiliary theorem and the proof of, 4 Proof of, it develops the needed tools to prove Lemma 4.4 and Theorem 3.2. We shall use the notation in Definition 4.2 and develop some more tools. For sets X and Y, the symmetric difference of X and Y is XΔY=(XY)(XY). If an edge e=uvE(G) but u,vV(G), then let G+e be the graph containing G as a spanning subgraph with edge set E(G){e}. For vV(G) and eE(G), we first study reduced

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (Nos. 11701490, 11771039, 11771443).

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