On -hamiltonicity of net-free line graphs
Introduction
We consider finite graphs without loops but permitting multiple edges, and follow [1] for undefined terms and notations. In particular, for a graph , , , and denote the connectivity, edge-connectivity, the minimum degree and the maximum degree of , respectively. We use and to denote the circumference and the girth of , which are the length of a longest cycle in and the length of a shortest cycle of , respectively. A graph is trivial if it has no edges. We write to mean that is a subgraph of . If , then is the subgraph of induced by . If and are subgraphs of a graph , then we define . Throughout this paper, we use to denote a path of order . For integers , let denote the graph formed by identifying each vertex of a with an end vertex of three disjoint paths , , of length , and , respectively. A graph is -free if contains no induced subgraph isomorphic to any copy of for any . If , then an -free graph is simply called an -free graph. A claw-free graph is just a -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 , denoted by , is a simple graph with vertex set , where two vertices in are adjacent if and only if the corresponding edges in 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 be a graph. (i) (Zhan, Theorem 3 in [37]) If , then 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 , as seen in the surveys in [2], [11], [13], [14], among others. The following have been proved.
Theorem 1.3 Let be the graph obtained from the Petersen graph by adding one pendant edge to each vertex. Let be a 3-connected simple claw-free graph. (i) (Brousek, Ryjáček and Favaron, [4]) If is -free, then is hamiltonian. (ii) [24] If is -free, then is hamiltonian. Moreover, the graph indicates the sharpness of this result. (iii) (Fujisawa, [12], see also Ma et al. [29]) If is -free graph, then is hamiltonian unless is the line graph of .
It is natural to seek necessary and sufficient conditions for hamiltonicity of line graphs. For an integer , a graph of order is -hamiltonian (-Hamilton-connected, respectively), if for any with , is hamiltonian ( is Hamilton-connected, respectively). It is well known that if a graph is s-hamiltonian, then is -connected, and if is s-Hamilton-connected, then is -connected. Broersma and Veldman in [3] initiated the problem of investigating graphs whose line graph is -hamiltonian if and only if the connectivity of the line graph is at least . They define, for an integer , a graph to be -triangular if every edge of lies in at least triangles of . The following is obtained.
Theorem 1.4 Broersma and Veldman, [3] Let be integers and let be a -triangular simple graph. Then is -hamiltonian if and only is -connected.
Broersma and Veldman in [3] proposed an open problem of determining the range of an integer such that within triangular graphs, is -hamiltonian if and only is -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 and be positive integers such that , and let be a -triangular simple graph. Then is -hamiltonian if and only is -connected. (ii) [21] Let be a connected graph and let be an integer. Then is -hamiltonian if and only if is -connected.
An hourglass is a graph isomorphic to , where is a cycle of length 4 in . 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 , the line graph of a claw-free graph is -hamiltonian if and only if is -connected. (iii) [25] The line graph of a claw-free graph is -Hamilton-connected if and only if is -connected. (iv) (Hu and Zhang [16]) Every 3-connected -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 , is -hamiltonian if and only if . The main goal of this research is to investigate if in Theorem 1.3(ii) can be replaced by other and if further evidences to support the conjecture in [21] can be found. The following results are obtained.
Theorem 1.7 Let be an integer. (i) Let and . Then for any , every -free line graph with is -hamiltonian if and only if for . (ii) Let and . Then for any , every -free line graph with is -Hamilton-connected if and only if for .
Theorem 1.7 extends Theorem 1.3(i) in the context of line graph and furthers the main results in [36]. Let be the set of odd degree vertices of a graph . Following [1], a graph is eulerian if is connected with . A graph is supereulerian if 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 be a 2-edge-connected graph. Each of the following holds. (i) Let be a graph with and . If and , then is supereulerian. (ii) If and has at most two edge-cuts of size 2, then 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 is collapsible if for every subset with (mod 2), has a spanning connected subgraph such that . See Catlin’s survey [7] and it supplements [8], [22] for further literature in this area. We use the notation that for a graph and an integer , define .
For a graph and , the contraction is the graph formed from by contracting edges in 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 . 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 be a connected graph with a 2-edge-cut and let and be the two components of . If both and are supereulerian, then is also supereulerian.
Proof For , let denote the vertex in onto which 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 , we use to denote the cyclic group of order . For integers , let be the graph obtained from disjoint paths and by identifying an end vertex of each of these three paths. (See Fig. 1 in [36] for an example.) By definition, . Define 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 and , the symmetric difference of and is . If an edge but , then let be the graph containing as a spanning subgraph with edge set . For and , 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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