The reliability polynomial of a graph gives the probability that remains operational when all its edges could fail independently with certain fixed probability. In general, problem finding uniformly most reliable graphs inside family graphs, is, one whose is at least as large any other family, very difficult. this paper, we study in containing hamiltonian cycle.

We consider the pair of graphs S and T such that every 3-connected {S, T}-free graph is hamiltonian connected. Such pairs are called hamiltonian connected pairs. We show three new hamiltonian connected pairs. Combing our results and a result of Shepherd, we conclude that if {S, T} is a pair of graphs such that every 2-connected {S, T}-free graph is hamiltonian then every 3-connected {S, T}-free...

Grinberg’s theorem is a necessary condition for the planar Hamilton graphs. In this paper, we use cycle bases and removable cycles to survey cycle structures of the Hamiltonian graphs and derive an equation of the interior faces in Grinberg’s Theorem. The result shows that Grinberg’s Theorem is suitable for the connected and simple graphs. Furthermore, by adding a new constraint of solutions to...

A graph G is hypohamiltonian/hypotraceable if it is not hamiltonian/traceable, but all vertex deleted subgraphs of G are hamiltonian/traceable. Until now all hypotraceable graphs were constructed using hypohamiltonian graphs; extending a method of Thomassen [8] we present a construction that uses so-called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex deleted subgraphs are ...

A graph G is called hamiltonian-<:onnectedfrom a vertex v ifa hamiltonian path exists from v to every other vertex w ~ v. We present a survey of the main results known about such graphs, including a section on graphs uniquely hamiltonian-connected from a venex and a section on the computational complexity ofdetermining whether a given graph is hamiltonian-<:onnectedfrom a venex or uniquely hami...

In a previous paper the authors showed that almost all labelled cubic graphs are hamiltonian. In the present paper, this result is used to show that almost all r-regular graphs are hamiltonian for any fixed r ≥ 3, by an analysis of the distribution of 1-factors in random regular graphs. Moreover, almost all such graphs are r-edge-colourable if they have an even number of vertices. Similarly, al...