Characterizing 4-critical graphs with Ore-degree at most seven

Abstract

Dirac introduced the notion of a k-critical graph, a graph that is not (k−1)-colorable but whose every proper subgraph is (k−1)-colorable. Brook's Theorem states that every graph with maximum degree k is k-colorable unless it contains a subgraph isomorphic to Kk+1 or an odd cycle (for k=2). Equivalently, for all k≥4, the only k-critical graph of maximum degree k−1 is Kk. A natural generalization of Brook's theorem is to consider the Ore-degree of a graph, which is the maximum of d(u)+d(v) over all uv∈E(G). Kierstead and Kostochka proved that for all k≥6 the only k-critical graph with Ore-degree at most 2k−1 is Kk. Kostochka, Rabern and Stiebitz proved that the only 5-critical graphs with Ore-degree at most 9 are K5 and a graph they called O5.A different generalization of Brook's theorem, motivated by Hajos' construction, is Ore's conjectured bound on the minimum density of a k-critical graph. Recently, Kostochka and Yancey proved Gallai's conjecture. Their proof for k≥5 implies the above results on Ore-degree. However, the case for k=4 remains open, which is the subject of this paper.Kostochka and Yancey's short but beautiful proof for the case k=4 says that if G is a 4-critical graph, then |E(G)|≥(5|V(G)|−2)/3. We prove the following bound which is better when there exists a large independent set of degree three vertices: if G is a 4-critical graph G, then |E(G)|≥1.6|V(G)|+.2α(D3(G))−.6, where D3(G) is the graph induced by the degree three vertices of G. As a corollary, we characterize the 4-critical graphs with Ore-degree at most seven as precisely the graphs of Ore-degree seven in the family of graphs obtained from K4 and Ore compositions.