About chromatic uniqueness of some complete tripartite graphs

TL;DR

This paper proves that certain complete tripartite graphs are uniquely identified by their chromatic polynomial under specific size constraints, contributing to the understanding of graph chromatic uniqueness.

Contribution

It establishes conditions under which full tripartite graphs are chromatically unique, extending previous knowledge in graph coloring theory.

Findings

Full tripartite graphs with specified size constraints are chromatically unique.

Chromatic polynomial uniquely determines these graphs up to isomorphism.

Results depend on inequalities involving the part sizes and modular conditions.

Abstract

Let $P(G, x)$ be the chromatic polynomial of a graph $G$. A graph $G$ is called \textit{chromatically unique} if for any graph $H,\, P(G, x) = P(H, x)$ implies that $G$ and $H$ are isomorphic. In this paper we show that full tripartite graph $K(n_1, n_2, n_3)$ is chromatically unique if $n_1 \geq n_2 \geq n_2 \geq n_3 \geq 2, n_1 - n_3 \leq 5$ and $n_1 + n_2 + n_3 \not \equiv 2 \mod{3}$.