A Note on The Vertex-Distinguishing Proper Total Coloring of Graphs.

Abstract

Let $G$ be a simple graph of order $p\ge 2$. A proper $k$-total coloring of a simple graph $G$ is called a $k$-vertex distinguishing proper total coloring ($k$-VDTC) if for any two distinct vertices $u$ and $v$ of $G$, the set of colors assigned to $u$ and its incident edges differs from the set of colors assigned to $v$ and its incident edges. The notation ${\chi }_{vt}(G)$ indicates the smallest number of colors required for which $G$ admits a $k$-VDTC with $k\ge {\chi }_{vt}(G)$. For every integer $m\ge 3$, we will present a graph $G$ of maximum degree $m$ such that ${\chi }_{vt}(G)<{\chi }_{vt}(H)$ for some proper subgraph $H\subseteq G$.