On Adjacent Vertex-distinguishing Total Chromatic Number of Generalized Mycielski Graphs

Abstract

The adjacent vertex-distinguishing total chromatic number of a graph $G$, denoted by $\chi_{at}(G)$, is the smallest $k$ for which $G$ has a proper total $k$-coloring such that any two adjacent vertices have distinct sets of colors appearing on the vertex and its incident edges. In regard of this number, there is a famous conjecture (AVDTCC) which states that for any simple graph $G$, $\chi_{at}(G) \leq \Delta(G)+3$. In this paper, we study this number for the generalized Mycielski graph $\mu_m(G)$ of a graph $G$. We prove that the satisfiability of the conjecture AVDTCC in $G$ implies its satisfiability in $\mu_m(G)$. Particularly we give the exact values of $\chi_{at}(\mu_m(G))$ when $G$ is a graph with maximum degree less than $3$ or a complete graph. Moreover, we investigate $\chi_{at}(G)$ for any graph $G$ with only one maximum degree vertex by showing that $\chi_{at}(G) \leq \Delta(G)+2$ when $\Delta(G) \leq 4$.