Coloring Graphs with Forbidden Minors
Martin Rolek, Zi-Xia Song

TL;DR
This paper advances the understanding of graph coloring related to forbidden minors, providing new bounds and proofs for graphs excluding certain complete minors, and introduces a novel Kempe-chain method.
Contribution
It offers a short, computer-free proof for graphs with no $K_t$ minor being $(2t-6)$-colorable for specific $t$, and generalizes this result with new bounds and methods.
Findings
Graphs with no $K_7$ minor are $6$-colorable.
Graphs with no $K_8^-$ minor are $9$-colorable.
Kempe-chain method developed is of independent interest.
Abstract
Hadwiger's conjecture from 1943 states that for every integer , every graph either can be -colored or has a subgraph that can be contracted to the complete graph on vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no minor are -colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no minor are -colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence number two, we first give a very short and computer-free proof of a recent result of Albar and Gon\c{c}alves and generalize it to the next step by showing that every graph with no minor is -colorable, where . We then prove that graphs with no minor are -colorable…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph theory and applications
