The adjacent vertex distinguishing total chromatic number

TL;DR

This paper investigates a strengthened total graph coloring concept where adjacent vertices have distinct incident color sets, establishing an upper bound of ( + C) colors for any graph.

Contribution

It introduces a new variant of total coloring ensuring incident color sets differ for adjacent vertices and proves a universal upper bound related to maximum degree.

Findings

Existence of such colorings with at most ( + C) colors

Universal constant C independent of the graph

Strengthening of total chromatic number concept

Abstract

A well-studied concept is that of the total chromatic number. A proper total colouring of a graph is a colouring of both vertices and edges so that every pair of adjacent vertices receive different colours, every pair of adjacent edges receive different colours and every vertex and incident edge receive different colours. This paper considers a strengthening of this condition and examines the minimum number of colours required for a total colouring with the additional property that for any adjacent vertices $u$ and $v$, the sets of colours incident to $u$ is different from the set of colours incident to $v$. It is shown that there is a constant $C$ so that for any graph $G$, there exists such a colouring using at most $\Delta(G) + C$ colours.