Author:
Albertson Michael O.,Cranston Daniel W.,Fox Jacob
Abstract
Albertson conjectured that if graph $G$ has chromatic number $r$, then the crossing number of $G$ is at least that of the complete graph $K_r$. This conjecture in the case $r=5$ is equivalent to the four color theorem. It was verified for $r=6$ by Oporowski and Zhao. In this paper, we prove the conjecture for $7 \leq r \leq 12$ using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds by Pach et al. on the crossing number of graphs in terms of the number of edges and vertices.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
4 articles.
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