Heavy subgraphs, stability and Hamiltonicity

Abstract

Let G be a graph. Adopting the terminology of Broersma et al. andČada, respectively, we say that G is 2-heavy if every induced claw (K 1,3 ) of G contains two end-vertices each one has degree at least |V (G)|/2; and G is o-heavy if every induced claw of G contains two end-vertices with degree sum at least |V (G)| in G. In this paper, we introduce a new concept, and say that G is S-c-heavy if for a given graph S and every induced subgraph G ′ of G isomorphic to S and every maximal clique C of G ′ , every non-trivial component of G ′ − C contains a vertex of degree at least |V (G)|/2 in G. In terms of this concept, our original motivation that a theorem of Hu in 1999 can be stated as every 2-connected 2-heavy and N -c-heavy graph is hamiltonian, where N is the graph obtained from a triangle by adding three disjoint pendant edges. In this paper, we will characterize all connected graphs S such that every 2-connected o-heavy and S-c-heavy graph is hamiltonian. Our work results in a different proof of a stronger version of Hu's theorem. Furthermore, our main result improves or extends several previous results.

On implicit heavy subgraphs and Hamiltonicity of 2-connected graphs

“…Theorem 8 (Li and Ning [23]). Let S be a connected graph of order at least three and let G be a 2-connected claw-o-heavy graph.…”

“…Recently, motivated by the main result of [20], Li and Ning [23] introduced another type of heavy subgraphs. We say that an induced subgraph H of G is c-heavy in G, if for every maximal clique C of H every non-trivial component of H − C contains a vertex that is heavy in G. Graph G is said to be H-c-heavy if every induced subgraph of G isomorphic to H is c-heavy.…”

“…Observe that every graph is trivially {K 1,3 , C 3 , P 3 }-c-heavy, since removal of a maximal clique from any of the three subgraphs results in a graph consisting of trivial components (or an empty graph). With that remark in mind, the authors of [23] extended Theorem 2 in the following way.…”

Degree, toughness and subgraph conditions for hamiltonian properties of graphs

“…The earliest ideas in this direction date back to the 1990s by accounts in [19] and [17]. These ideas gave rise to the notion of a heavy subgraph, and according to the different requirements of the degree on subgraphs people studied different notions, such as the notions of an f-heavy [68], o-heavy [21,68] and c-heavy subgraph [55]. Just to give the flavor of these results, without going into detail and without defining the special graphs that appear in the below statements, we mention a few examples of results in which different groups of authors have fully characterized the pairs that imply hamiltonicity.…”

“…Then G being {R, S}-f-heavy implies that G is hamiltonian if and only if (up to symmetry) R = K 1,3 and S is one of the following: -P 4 , P 5 , P 6 (Chen, Wei and Zhang [29]), -Z 1 (Bedrossian, Chen and Schelp [7]), -B (Li, Wei and Gao [58]), -N (Chen, Wei and Zhang [27]), -Z 2 , W (Ning and Zhang [68]). [55]). Let S be a connected graph of order at least three and let G be a 2-connected claw-o-heavy graph.…”