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

A set SS of vertices in a graph GG is a total dominating set of GG if every vertex of GG is adjacent to some vertex in SS. The minimum cardinality of a total dominating set of GG is the total domination number γt(G)γt(G). The graph GG is 3t3t-critical if γt(G)=3γt(G)=3 and γt(G+e)=2γt(G+e)=2 for every edge ee in the complement of GG. We show that no bipartite graph is 3t3t-critical. The tripartite 3t3t-critical graphs are characterized. For every k≥3k≥3, we prove that there are only a finite number of 3t3t-critical kk-partite graphs. We show that the 55-cycle is the only 3t3t-critical K3K3-free graph and that there are only a finite number of 3t3t-critical K4K4-free graphs.