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

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 $χ_{r-vsdt}(G)$, is the minimum integer $k$ for which $G$ admits a $k$-$D(r)$-VSDTC. We show that $χ_{1-vsdt}(G)\leq 4Δ(G)$ for every graph $G$ without isolated edges and $χ_{1-vsdt}(G)\le kΔ(G)+3$ for a $k$-degenerated graph $G$ without isolated edges, where $1\le k\le 3$.

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