Complete and almost complete minors in double-critical 8-chromatic graphs
The electronic journal of combinatorics, Tome 18 (2011) no. 1
A connected $k$-chromatic graph $G$ is said to be 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 [Electron. J. Combin., 17(1): Research Paper 87, 2010] proved that every double-critical $k$-chromatic graph with $k \leq 7$ contains a $K_k$ minor. It remains unknown whether an arbitrary double-critical $8$-chromatic graph contains a $K_8$ minor, but in this paper we prove that any double-critical $8$-chromatic contains a minor isomorphic to $K_8$ with at most one edge missing. In addition, we observe that any double-critical $8$-chromatic graph with minimum degree different from $10$ and $11$ contains a $K_8$ minor.
@article{10_37236_567,
author = {Anders Sune Pedersen},
title = {Complete and almost complete minors in double-critical 8-chromatic graphs},
journal = {The electronic journal of combinatorics},
year = {2011},
volume = {18},
number = {1},
doi = {10.37236/567},
zbl = {1218.05052},
url = {http://geodesic.mathdoc.fr/articles/10.37236/567/}
}
Anders Sune Pedersen. Complete and almost complete minors in double-critical 8-chromatic graphs. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/567
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