A Hamiltonian cycle in a graph G is a simple cycle in which each vertex of G appears exactly once. The Hamiltonian cycle problem involves testing whether a Hamiltonian cycle exists in a graph, and finds one if such a cycle does exist. It is well known that the Hamiltonian cycle problem is one of the classic NP-complete problems on general graphs. Shih et al. solved the Hamiltonian cycle problem...

A cycle C of length k is extendable if there is a cycle C′ of length k+1 with V (C) ⊂ V (C′). A graph G = (V,E) of order n is cycle extendable when every cycle C of length k < n is extendable. A chordal graph is a spider intersection graph if it admits an intersection representation which consists of subtrees of a sub-divided star (or spider). In 1990, Hendry conjectured that all hamiltonian ch...

In a finite undirected simple graph, a chordless cycle is an induced subgraph which is a cycle. A graph is called cyclically orientable if it admits an orientation in which every chordless cycle is cyclically oriented. We propose an algorithm to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithm have the advantage of finding each chordle...

A cycle C of a graph G is dominating if each component of GnC is edgeless. In the paper, we will give two sufficient conditions for each longest cycle of a 3-connected graph to be a dominating cycle. 2005 Wiley Periodicals, Inc. J Graph Theory 49: 135–150, 2005

In this paper, we have shown that collection of paths P i n where n is odd, cycle, triangular snake, quadrilateral snake, P 2 n are even sequential harmonious graph and also we prove that the cycle Cn be a cycle u1, u2, · · · , un is an even sequential harmonious graph. Let G be a graph with V (G) = V (Cn) ∪ {wi : 1 ≤ i ≤ n}, E(G) = E(Cn) ∪ {uiwiui−1wi : 1 ≤ i ≤ n}, then G is an even sequential...

In this note we show that deciding the existence of a Hamiltonian cycle in a cubic plane graph is equivalent to the problem of the existence of an associated cubic plane multi-3-gonal graph with a Hamiltonian cycle which takes alternately left and right edges at each successive vertex, i.e. it is also a Petrie cycle. The Petrie Hamiltonian cycle in an n-vertex plane cubic graph can be recognize...

Let G = (V, E) be a simple connected graph, then the basis number of is denoted by b(G) and defined least positive integer k such that graph has − fold for it cycle space. In this paper we studied Mycielski’s some cog-special graphs, compute cog-path cog-cycle cog-star cog-wheel graph.

A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K„), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices . Then, another cycle-complete graph Ramsey number is studied, namely r( :C,,,, K„) the smallest order for a graph which forces it to contain either a cycle of order 1 for some I satisfying 3 :1...

Ng and Schultz [J Graph Theory 1 (1997), 45±57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k-ordered Hamiltonian. We give sum of degree conditions for nonadja...

In this paper, we continue the study of the domination game in graphs introduced by Bre{v{s}}ar, Klav{v{z}}ar, and Rall. We study the paired-domination version of the domination game which adds a matching dimension to the game. This game is played on a graph $G$ by two players, named Dominator and Pairer. They alternately take turns choosing vertices of $G$ such that each vertex chosen by Domin...