Hamiltonicity of 3tEC Graphs with α=κ+1

Abstract

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γtG of G. The graph G is total domination edge-critical, or γtEC, if for every edge e in the complement of G, γtG+e<γtG. If G is γtEC and γtG=k, we say that G is ktEC. In this paper, we show that every 3tEC graph with δG≥2 and αG=κG+1 has a Hamilton cycle.