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

Subgraphs, Closures and Hamiltonicity

Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c(u,v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c(u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G

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