Abstract

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypothesis that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots,X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.

Keywords

  1. improper coloring
  2. minor
  3. Hadwiger's conjecture

MSC codes

  1. 05C15
  2. 05C83

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 2385 - 2388
ISSN (online): 1095-7146

History

Submitted: 31 December 2014
Accepted: 14 October 2015
Published online: 10 December 2015

Keywords

  1. improper coloring
  2. minor
  3. Hadwiger's conjecture

MSC codes

  1. 05C15
  2. 05C83

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