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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 graphsmeeting 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 ofvertex-disjoint cliques of size three or four where cliques of size four havetwice the weight of a clique of size three. As a corollary, a triangle-free5-critical graph G satisfies: |E(G)|>=(9/4 + e)|V(G)| - 5/4.

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