On the existence of -partite or -free total domination edge-critical graphs

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 γ t ( G ) . The graph G is 3 t -critical if γ t ( G ) = 3 and γ t ( G + e ) = 2 for every edge e in the complement of G . We show that no bipartite graph is 3 t -critical. The tripartite 3 t -critical graphs are characterized. For every k ≥ 3 , we prove that there are only a finite number of 3 t -critical k -partite graphs. We show that the 5 -cycle is the only 3 t -critical K 3 -free graph and that there are only a finite number of 3 t -critical K 4 -free graphs.