Long cycles in graphs with large degree sums and neighborhood unions

We present and prove several results concerning the length of longest cycles in 2-connected or 1-tough graphs with large degree sums. These results improve many known results on long cycles in these graphs. We also consider the sharpness of the results and discuss some possible strengthenings

Long cycles, degree sums and neighborhood unions

AbstractFor a graph G, define the parameters α(G)=max{|S| |S is an independent set of vertices of G}, σk(G)=min{∑ki=1d(vi)|{v1,…,vk} is an independent set} and NCk(G)= min{|∪ki=1 N(vi)∥{v1,…,vk} is an independent set} (k⩾2). It is shown that every 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,n+NCr+5+∈(n+r)(G)-α(G)}, where ε(i)=3(⌈13i⌉−13i). This result extends previous results in Bauer et al. (1989/90), Faßbender (1992) and Flandrin et al. (1991). It is also shown that a 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,2NC⌊18(n+6r+17)⌋(G)}. Analogous results are established for 2-connected graphs

A generalization of Ore's Theorem involving neighborhood unions

AbstractLet G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-connected and |N(u)∪N(v)|⩾12n for every pair of nonadjacent vertices u,v, then either G is hamiltonian, or G is the Petersen graph, or G belongs to one of three families of exceptional graphs of connectivity 2

A note on dominating cycles in 2-connected graphs

Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs

A closure concept based on neighborhood unions of independent triples

The well-known closure concept of Bondy and Chvatal is based on degree-sums of pairs of nonadjacent (independent) vertices. We show that a more general concept due to Ainouche and Christofides can be restated in terms of degree-sums of independent triples. We introduce a closure concept which is based on neighborhood unions of independent triples and which also generalizes the closure concept of Bondy and Chvatal

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Hamiltonian properties of graphs with large neighborhood unions

AbstractLet G be a graph of order n, σk = min{ϵi=1kd(νi): {ν1,…, νk} is an independent set of vertices in G}, NC = min{|N(u)∪ N(ν)|: uν∉E(G)} and NC2 = min{|N(u)∪N(ν)|: d(u,ν)=2}. Ore proved that G is hamiltonian if σ2⩾n⩾3, while Faudree et al. proved that G is hamiltonian if G is 2-connected and NC⩾13(2n−1). It is shown that both results are generalized by a recent result of Bauer et al. Various other existing results in hamiltonian graph theory involving degree-sums or cardinalities of neighborhood unions are also compared in terms of generality. Furthermore, some new results are proved. In particular, it is shown that the bound 13(2n−1) on NC in the result of Faudree et al. can be lowered to 13(2n−1), which is best possible. Also, G is shown to have a cycle of length at least min{n, 2(NC2)} if G is 2-connected and σ3⩾n+2. A Dλ-cycle (Dλ-path) of G is a cycle (path) C such that every component of G−V(C) has order smaller than λ. Sufficient conditions of Lindquester for the existence of Hamilton cycles and paths involving NC2 are extended to Dλ-cycles and Dλ-paths