Low Diameter Monochromatic Covers of Complete Multipartite Graphs

TL;DR

This paper investigates the minimum diameter needed for two monochromatic subgraphs to cover all vertices in any two-color edge coloring of complete multipartite graphs, providing exact values for tripartite graphs and nearly all with more parts.

Contribution

It precisely determines the diameter cover number for two monochromatic subgraphs in complete tripartite graphs and almost all larger multipartite graphs under two-colorings.

Findings

Exact value of D_2^2(G) for all complete tripartite graphs.

Almost complete characterization of D_2^2(G) for multipartite graphs with more than three parts.

Provides a comprehensive understanding of monochromatic covers in multipartite graphs.

Abstract

Let the diameter cover number, $D^t_r(G)$, denote the least integer $d$ such that under any $r$-coloring of the edges of the graph $G$, there exists a collection of $t$ monochromatic subgraphs of diameter at most $d$ such that every vertex of $G$ is contained in at least one of the subgraphs. We explore the diameter cover number with two colors and two subgraphs when $G$ is a complete multipartite graph with at least three parts. We determine exactly the value of $D_2^2(G)$ for all complete tripartite graphs $G$, and almost all complete multipartite graphs with more than three parts.