Most direct product of graphs are Type 1

TL;DR

This paper proves that most direct product graphs of cycles are Type 1, confirming a long-standing conjecture except for the specific case of C4×C4, and explores conditions for graphs to reach the lower total chromatic number bound.

Contribution

The paper establishes that all cycle graph direct products C_m×C_n are Type 1 except for C4×C4, confirming a significant conjecture in total graph coloring.

Findings

All C_m×C_n are Type 1 except C4×C4.

Confirmed the conjecture for a broad class of cycle graphs.

Provided conditions for graphs to attain the lower total chromatic number.

Abstract

A \textit{$k$-total coloring} of a graph $G$ is an assignment of $k$ colors to its elements (vertices and edges) so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer $k$ for which the graph $G$ has a $k$-total coloring. Clearly, this number is at least $\Delta(G)+1$, where $\Delta(G)$ is the maximum degree of $G$. When the lower bound is reached, the graph is said to be Type~1. The upper bound of $\Delta(G)+2$ is a central problem that has been open for fifty years, is verified for graphs with maximum degree 4 but not for regular graphs. Most classified direct product of graphs are Type~1. The particular cases of the direct product of cycle graphs $C_m \times C_n$, for $m =3p, 5\ell$ and $8\ell$ with $p \geq 2$ and $\ell \geq 1$, and arbitrary $n \geq 3$, were previously known to be Type 1 and motivated the conjecture that, except for $C_4 \times C_4$, all direct product of cycle graphs $C_m \times C_n$ with $m,n \geq 3$ are Type 1. We give a general pattern proving that all $C_m \times C_n$ are Type 1, except for $C_4 \times C_4$. dditionally, we investigate sufficient conditions to ensure that the direct product reaches the lower bound for the total chromatic number.