Complete and almost complete minors in double-critical 8-chromatic graphs

2010/07/30 by Anders Sune Pedersen, Pedersen, Anders Sune
Mathematics · #05C15 #05C69 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C15 #msc:05C69

paper · pdf · doi:10.48550/arxiv.1007.5400

arxiv created 2010/07/30 · arxiv updated 2010/08/02

Abstract

A connected k-chromatic graph G is said to be \it double-critical if for all edges uv of G the graph G - u - v is (k-2)-colourable. A longstanding conjecture of Erdős and Lovász states that the complete graphs are the only double-critical graphs. Kawarabayashi, Pedersen and Toft [\itElectron. J. Combin., 17(1): Research Paper 87, 2010] proved that every double-critical k-chromatic graph with k ≤ 7 contains a K_k minor. It remains unknown whether an arbitrary double-critical 8-chromatic graph contains a K₈ minor, but in this paper we prove that any double-critical 8-chromatic contains a K₈^- minor; here K₈^- denotes the complete 8-graph with one edge missing. In addition, we observe that any double-critical 8-chromatic graph with minimum degree different from 10 and 11 contains a K₈ minor.

Related

• Double-critical graphs and complete minors
• Clique Minors in Double-critical Graphs
• A note on the double-critical graph conjecture
• Double-critical graph conjecture for claw-free graphs
• Properties of 8-contraction-critical graphs with no K₇ minor