On the Minimum Edge-Density of 5-Critical Triangle-Free Graphs

Kostochka and Yancey proved that every 5-critical graph G satisfies: . A construction of Ore gives an infinite family of graphs meeting this bound.We prove that there exists ϵ,δ>0 such that if G is a 5-critical graph, then 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: .