The competition number of a graph and the dimension of its hole space

TL;DR

This paper explores the relationship between a graph's competition number and the dimension of its hole space, proposing a conjecture and verifying it for specific graph classes, especially triangle-free graphs.

Contribution

It introduces a conjecture linking competition number and hole space dimension, and proves the equality for connected triangle-free graphs.

Findings

The conjecture that the hole space dimension is at least the competition number is supported.

Equality holds for connected triangle-free graphs.

The relationship provides insights into graph characterization by these parameters.

Abstract

The competition graph of a digraph D is a (simple undirected) 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 G is 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 to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph is being studied. A hole of a graph is a cycle of length at least 4 as an induced subgraph. In this paper, we conjecture that the dimension of the hole space of a graph is no smaller than the competition number of the graph. We verify this conjecture for various kinds of graphs and show that our conjectured inequality is indeed an equality for connected triangle-free graphs.