Characterizing 4-Critical Graphs of Ore-Degree at most Seven

TL;DR

This paper investigates 4-critical graphs with bounded Ore-degree, providing a new lower bound on edges involving degree-three vertices, and characterizes those with Ore-degree at most seven.

Contribution

It introduces an improved edge bound for 4-critical graphs based on degree-three vertices and characterizes graphs with Ore-degree at most seven.

Findings

Established a new edge bound involving degree-three vertices.

Characterized 4-critical graphs with Ore-degree at most seven.

Connected the structure of these graphs to Ore compositions.

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 K_{k+1} (or an odd cycle for k=2). Equivalently, for all k>=4, the only k-critical graph of maximum degree k-1 is K_k. 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 edges uv. Kierstead and Kostochka proved that for all k>=6 the only k-critical graph with Ore-degree at most 2k-1 is K_k. Kostochka, Rabern and Steibitz proved that the only 5-critical graphs with Ore-degree at most 9 are K_5 and a graph they called O_5. A different generalization of Brook's theorem, motivated by Hajos' construction, is Gallai'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)| + .2 alpha(D_3(G)) - .6, where D_3(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 K_4 and Ore compositions.