New Ore’s Type Results on Hamiltonicity and Existence of Paths of Given Length in Graphs

Abstract

The well-known Ore’s theorem (see Ore in Am Math Mon 65:55, 1960), states that a graph G of order n such that d(x) + d(y) ≥ n for every pair {x, y} of non-adjacent vertices of G is Hamiltonian. In this paper, we considerably improve this theorem by proving that in a graph G of order n and of minimum degree δ ≥ 2, if there exist at least n − δ vertices x of G so that the number of the vertices y of G non-adjacent to x and satisfying d(x) + d(y) ≤ n − 1 is at most δ − 1, then G is Hamiltonian. We will see that there are graphs which violate the condition of the so called “Extended Ore’s theorem” (see Faudree et al. in Discrete Math 307:873–877, 2007) as well as the condition of Chvatál’s theorem (see Chvátal in J Combin Theory Ser B 12:163–168, 1972) and the condition of the so called “Extended Fan’ theorem” (see Faudree et al. in Discrete Math 307:873–877, 2007), but satisfy the condition of our result, which then, only allows to conclude that such graphs are Hamiltonian. This will show the pertinence of our result. We give also a new result of the same type, ensuring the existence of a path of given length.