On the Minimum Number of Edges in Triangle-Free 5-Critical Graphs

TL;DR

This paper improves the lower bound on the number of edges in 5-critical graphs, especially triangle-free ones, by introducing additional parameters and proving a stronger inequality than previous results.

Contribution

It establishes a new lower bound on edges in 5-critical graphs involving a parameter related to vertex-disjoint cliques, refining prior bounds.

Findings

New lower bound on edges in 5-critical graphs

Stronger bound for triangle-free 5-critical graphs

Introduction of a parameter T(G) related to vertex-disjoint cliques

Abstract

Kostochka and Yancey proved that every 5-critical graph G satisfies: |E(G)|>= (9/4)|V(G)| - 5/4. A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists e,d > 0 such that if G is a 5-critical graph, then |E(G)| >= (9/4 + e)|V(G)|- 5/4 - dT(G), where T(G) is the maximum number of vertex-disjoint cliques of size three or four where cliques of size four have twice the weight of a clique of size three. As a corollary, a triangle-free 5-critical graph G satisfies: |E(G)|>=(9/4 + e)|V(G)| - 5/4.