On the minimum number of edges in triangle-free 5-critical graphs

Kostochka and Yancey proved that every 5-critical graph G satisfies: vertical bar E(G)vertical bar >= 9/4 vertical bar V(G)vertical bar - 5/4. A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists epsilon, delta > 0 such that if G is a 5-critical graph, then vertical bar E(G)vertical bar >= (9/4 + epsilon)vertical bar V(G)vertical bar - 5/4 - delta T(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: vertical bar E(G)vertical bar >= (9/4 + epsilon)vertical bar V(G)vertical bar - 5/4. (C) 2017 Elsevier Ltd. All rights reserved.