Upper Bounds on the $D(\beta )$-Vertex- Distinguishing Total-Chromatic Number of Graphs

Abstract

In this paper, we study the upper bounds for the $D(\beta )$-vertex-distinguishing total-chromatic numbers using the probability method, and obtain: Let $\mathrm{\Delta }$ be the maximum degree of $G$, then

$${\chi }_{\beta vt}\le \{\begin{array}{ll}16{\mathrm{\Delta }}^{(\beta +1)/(2\mathrm{\Delta }+2)},& \mathrm{\Delta }\ge 3,\beta \ge 4\mathrm{\Delta }+3;\\ 13{\mathrm{\Delta }}^{(\beta +4)/4},& \mathrm{\Delta }\ge 4,\beta \ge 5;\\ 10{\mathrm{\Delta }}^2,& \mathrm{\Delta }\ge 3,2\le \beta \le 4.\end{array}$$