A property P defined on all graphs of order n is said to be k-stable if for any graph of order n that does not satisfy P , the fact that uv is not an edge of G and that G+uv satisfies P implies dG(u)+dG(v) < k. Every property is (2n−3)-stable and every k-stable property is (k+1)stable. We denote by s(P ) the smallest integer k such that P is k-stable and call it the stability of P . This number...

A graph G on n vertices is said to be (k, m)-pancyclic if each k-set S ⊆ V (G) is contained in cycles of each of the following lengths: m,m + 1, · · · , n. This property generalizes vertex pancyclicity. There are ten pairs of forbidden subgraphs which guarantee that a 2-connected graph is (k,m)-pancyclic for some integer m ≤ n. We give the best (smallest) possible value for m in each of these t...

In this paper, we deal with the graph G0⊕G1 obtained from merging two graphs G0 and G1 with n vertices each by n pairwise nonadjacent edges joining vertices in G0 and vertices in G1. The main problems studied are how fault-panconnectivity and fault-pancyclicity of G0 and G1 are translated into fault-panconnectivity and fault-pancyclicity of G0⊕G1, respectively. Many interconnection networks suc...

In this paper, we present several conditions for K1;3-free graphs, which guarantee the graph is subpancyclic. In particular, we show that every K1;3-free graph with a minimum degree sum 2 ¿ 2 √ 3n+ 1 − 4; every {K1;3; P7}-free graph with 2 ¿ 9; every {K1;3; Z4}-free graph with 2 ¿ 9; and every K1;3-free graph with maximum degree , diam(G)¡ ( +6)=4 and 2 ¿ 9 is subpancyclic. c © 2002 Elsevier Sc...