Equitable total coloring of corona of cubic graphs

TL;DR

This paper proves that the equitable total chromatic number of coronas of cubic graphs equals the maximum degree plus one, confirming the Total Coloring Conjecture and Equitable Total Coloring Conjecture for these graphs.

Contribution

It establishes that the equitable total chromatic number of coronas of cubic graphs is always maximum degree plus one, confirming two longstanding conjectures for this class of graphs.

Findings

Equitable total chromatic number of coronas of cubic graphs equals maximum degree plus one.

Confirms the Total Coloring Conjecture (TCC) for coronas of cubic graphs.

Confirms the Equitable Total Coloring Conjecture (ETCC) for coronas of cubic graphs.

Abstract

The minimum number of total independent partition sets of $V \cup E$ of a graph $G=(V,E)$ is called the \emph{total chromatic number} of $G$, denoted by $\chi''(G)$. If the difference between cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of $V \cup E$ is called the \emph{equitable total chromatic number}, and is denoted by $\chi''_=(G)$. In this paper we consider equitable total coloring of coronas of cubic graphs, $G \circ H$. It turns out that, independly on the values of equitable total chromatic number of factors $G$ and $H$, equitable total chromatic number of corona $G \circ H$ is equal to $\Delta(G \circ H) +1$. Thereby, we confirm Total Coloring Conjecture (TCC), posed by Behzad in 1964, and Equitable Total Coloring Conjecture (ETCC), posed by Wang in 2002, for coronas of cubic graphs. As a direct consequence we get that all coronas of cubic graphs are of Type 1.