Track D Social Science, Human Rights and Political Science

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/138414/1/jia218442.pd

Some localization theorems on hamiltonian circuits

Theorems on the localization of the conditions of G. A. Dirac (Proc. London Math. Soc. (3), 2 1952, 69–81), O. Ore (Amer. Math. Monthly, 67 1960, 55), and Geng-hua Fan (J. Combin. Theory Ser. B, 37 1984, 221–227) for a graph to be hamiltonian are obtained. It is proved, in particular, that a connected graph G on p ≥ 3 vertices is hamiltonian if d(u) ≥ | M3(u)|/2 for each vertex u in G, where M3(u) is the set of vertices v in G that are a distance at most three from u

Stable properties of graphs

Abstract For many properties P Bondy and Chvátal (1976) have found sufficient conditions such that if a graph G + uv has property P then G itself has property P. In this paper we will give a generalization that will improve ten of these conditions