On the existence problem of the total domination vertex critical graphs

The existence problem of the total domination vertex critical graphs has been studied in a series of articles. The aim of the present article is twofold. First, we settle the existence problem with respect to the parities of the total domination number m and the maximum degree Delta : for even m except m=4, there is no m-gamma_t-critical graph regardless of the parity of Delta; for m=4 or odd m \ge 3 and for even Delta, an m-gamma_t-critical graph exists if and only if Delta \ge 2 \lfloor \frac{m-1}{2}\rfloor; for m=4 or odd m \ge 3 and for odd Delta, if Delta \ge 2\lfloor \frac{m-1}{2}\rfloor +7, then m-gamma_t-critical graphs exist, if Delta < 2\lfloor \frac{m-1}{2}\rfloor, then m-gamma_t-critical graphs do not exist. The only remaining open cases are Delta = 2\lfloor \frac{m-1}{2}\rfloor +k, k=1, 3, 5. Second, we study these remaining open cases when m=4 or odd m \ge 9. As the previously known result for m = 3, we also show that for Delta(G)= 3, 5, 7, there is no 4-gamma_t-critical graph of order Delta(G)+4. On the contrary, it is shown that for odd m \ge 9 there exists an m-gamma_t-critical graph for all Delta \ge m-1.