$r$-strongly vertex-distinguishing total coloring of graphs

TL;DR

This paper introduces a new graph coloring parameter inspired by communication interference, establishing bounds on the minimum colors needed for graphs based on their maximum degree and degeneracy.

Contribution

It proposes the $r$-vertex-strongly-distinguishing total coloring and provides bounds on its chromatic number for specific classes of graphs.

Findings

Bound $ ext{for } r=1$, $ ext{chromatic number} \\leq 4\, ext{times maximum degree}$.

Bound for $k$-degenerated graphs: $ ext{chromatic number} \\leq k \, ext{times maximum degree} + 3$.

Applicable to graphs without isolated edges.

Abstract

Inspired by the phenomenon of co-channel interference in communication network, a novel graph parameter, called $r$-vertex-strongly-distinguishing total coloring (abbreviate as $D(r)$-VSDTC), is proposed in this paper. Given a graph $G$, an $r$-VSDTC is an assignment of $k$ colors to $V(G)\cup E(G)$ such that any two adjacent or incident elements receive different colors and any two vertices with distance at most $r$ have distinct color-set, where the color-set of a vertex $u$ is the set of colors assigned on $u$ and its neighborhoods and incident edges. The \emph{$r$-vertex-strongly-distinguishing total chromatic number} of $G$, denoted by $\chi_{r-vsdt}(G)$, is the minimum integer $k$ for which $G$ admits a $k$-$D(r)$-VSDTC. We show that $\chi_{1-vsdt}(G)\leq 4\Delta(G)$ for every graph $G$ without isolated edges and $\chi_{1-vsdt}(G)\le k\Delta(G)+3$ for a $k$-degenerated graph $G$ without isolated edges, where $1\le k\le 3$.