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 \(\Delta\) be the maximum degree of \(G\), then

\[
\chi_{\beta vt}\leq
\left\{
\begin{array}{ll}
16\Delta^{(\beta+1)/(2\Delta+2)}, & \Delta \geq 3,\beta\geq 4\Delta+3; \\
13\Delta^{(\beta+4)/4} , & \Delta\geq 4,\beta\geq 5;\\
10\Delta^2, & \Delta \geq 3, 2 \leq \beta \leq 4.
\end{array}
\right.
\]