An Implicit Degree Condition for Cyclability in Graphs

Abstract

A vertex subset X of a graph G is said to be cyclable in G if there is a cycle in G containing all vertices of X. Ore [6] showed that the vertex set of G with cardinality n ≥ 3 is cyclable (i.e. G is hamiltonian) if the degree sum of any pair of nonadjacent vertices in G is at least n. Shi [8] and Ota [7] respectively generalized Ore’s result by considering the cyclability of any vertex subset X of G under Ore type condition. Flandrin et al. [4] in 2005 extended Shi’s conclusion under the condition called regional Ore′s condition. Zhu, Li and Deng [10] introduced the definition of implicit degrees of vertices. In this work, we generalize the result of Flandrin et al. under their type condition with implicit degree sums. More precisely, we obtain that X is cyclable in a k-connected graph G if the implicit degree sum of any pair of nonadjacent vertices u,v ∈ X _i is at least the order of G, where each X _i , i = 1,2, ⋯ ,k is a vertex subset of G and X = ∪ ^k _i = 1 X _i . In [10], the authors demonstrated that the implicit degree of a vertex is at least the degree of the vertex. Hence our result is better than the result of Flandrin et al. in some way.