Adjacent Vertex Distinguishing Total Coloring of Corona Product of Graphs

TL;DR

This paper investigates a special type of graph coloring called adjacent vertex distinguishing total coloring, focusing on corona product graphs, and confirms a longstanding conjecture for many such graph classes.

Contribution

The paper proves the conjecture that every simple graph has an adjacent vertex distinguishing total coloring with at most 1(0(G)+3) colors for many corona product graphs, expanding the classes where the conjecture holds.

Findings

Confirmed the conjecture for many corona product graphs.

Established bounds for adjacent vertex distinguishing total coloring.

Extended results to generalized, simple, and l-coronas of graphs.

Abstract

An adjacent vertex distinguishing total $k$-coloring $f$ of a graph $G$ is a proper total $k$-coloring of $G$ such that no pair of adjacent vertices has the same color sets, where the color set at a vertex $v$, $C^G_f(v)$, is $\{f(v)\} \cup \{f(vu)|u \in V (G), vu \in E(G)\}$. In 2005 Zhang et al. posted the conjecture (AVDTCC) that every simple graph $G$ has adjacent vertex distinguishing total $(\Delta(G)+3)$-coloring. In this paper we confirm the conjecture for many coronas, in particular for generalized, simple and $l$-coronas of graphs, not relating the results to particular graph classes.