Hamiltonicity Of Total Domination Edge-critical Graphs With γt(G)=3

Domination theory is one of the signifcant areas in graph theory and the criticalproblem is the basis of the problem. Domination theory have many wide applications indiferent areas, communication network monitoring system, coding theory, social network,computer science and so on. The domination parameters meaning plays an importantrole in the structure of graphs. With the development of practical problems, the type setdomination parameters are increasing. In recent years, there are many research results indomination theory. These results not only promotes the development of graph theory, butalso is of great meaning in applying graph theory to practice. This paper mainly furtherstudies the critical problem of domination set of theory.A set S of vertices in a graphs G is a total dominating set of G if every vertex of G isadjacent to some vertex in S. The minimum cardinality of a total dominating set of G isthe total domination number γt(G) of G. The graphs G is total domination edge-critical,or γtEC, if for every edge e in the complement of G, γt(G+e)<γt(G). If G is γtECand γt(G)=k, we say that G is ktEC. In this paper, we show that if G is a3tECgraphs then α(G)≤κ(G)+2, with equality only if κ(G)=δ(G); every3tEC graph has aHamilton path and every3tEC graph with δ(G)≥2has a Hamilton cycle. We also posetwo related conjectures.