The competition numbers of Hamming graphs with diameter at most three

TL;DR

This paper determines the competition numbers of Hamming graphs with diameter up to three, addressing a complex problem in the study of competition graphs and providing exact values for these specific graphs.

Contribution

It explicitly computes the competition numbers for Hamming graphs with diameter at most three, a previously unresolved problem in graph theory.

Findings

Computed competition numbers for Hamming graphs with diameter ≤ 3

Provided exact values for these specific graphs

Advanced understanding of competition graph properties

Abstract

The competition graph of a digraph D is a graph which has the same vertex set as D and has an edge between x and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number k(G) for a graph G and it has been one of important research problems in the study of competition graphs. In this paper, we compute the competition numbers of Hamming graphs with diameter at most three.