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Hadwiger's conjecture for graphs with forbidden holes
Given a graph , the Hadwiger number of , denoted by , is the
largest integer such that contains the complete graph as a minor.
A hole in is an induced cycle of length at least four. Hadwiger's
Conjecture from 1943 states that for every graph , , where
denotes the chromatic number of . In this paper we establish more
evidence for Hadwiger's conjecture by showing that if a graph with
independence number has no hole of length between and
, then . We also prove that if a graph with
independence number has no hole of length between and
, then contains an odd clique minor of size , that is,
such a graph satisfies the odd Hadwiger's conjecture
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