On the 2-Vertex Fault Hamiltonicity for Graphs satisfying Ore's Theorem

TL;DR

This paper extends Ore's theorem by demonstrating that graphs satisfying it remain Hamiltonian after removing any two vertices, except for nine specific exceptional families.

Contribution

It generalizes previous results by proving the 2-vertex fault Hamiltonicity for graphs satisfying Ore's theorem, identifying nine exceptional graph families.

Findings

Graphs satisfying Ore's theorem are 2-vertex fault Hamiltonian.

Identifies nine exceptional families where 2-vertex fault Hamiltonicity does not hold.

Extends classical Hamiltonicity results to fault-tolerant scenarios.

Abstract

For any undirected and simple graph G = (V;E), where V denotes the vertex set and E the edge set of G. G is called hamiltonian if it contains a cycle that visits each vertex of G exactly once. Ore (1960) proved that G is hamiltonian if degG(u) + degG(v) >= n holds for every nonadjacent pair of vertices u and v in V, where n is the total number of distinct vertices of G. Kao et al. (2012) proved that any graph G satisfying Ore's theorem remains hamiltonian after the removal of any vertex x in V unless G belongs to one of the two exceptional families of graphs. In this paper, we proved that in fact, any graph satisfying Ore's theorem remains hamiltonian after the removal of two vertices x, y in V unless G belongs to one of the nine exceptional families of graphs.