On the Minimum Number of Edges in Triangle-Free 5-Critical Graphs
Combinatorics
2017-08-09 v2
Abstract
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 graphs meeting 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 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: |E(G)|>=(9/4 + e)|V(G)| - 5/4.
Keywords
Cite
@article{arxiv.1602.03098,
title = {On the Minimum Number of Edges in Triangle-Free 5-Critical Graphs},
author = {Luke Postle},
journal= {arXiv preprint arXiv:1602.03098},
year = {2017}
}
Comments
25 pages, revised according to referee comments