Dominating Hadwiger’s Conjecture

A dominating Kt minor in a graph G is a sequence (T1, ...,
Tt) of pairwise disjoint non-empty connected subgraphs of G, such that
for 1 ≤ i < j ≤ t, every vertex in Tj has a neighbor in Ti.
Replacing “every vertex in Tj” by “some vertex in Tj” retrieves the
standard definition of a Kt minor. The strengthened notion was
introduced by Illingworth and Wood (arXiv:2405.14299), who asked
whether every graph with chromatic number t contains a dominating Kt
minor. This is a substantial strengthening of the celebrated
Hadwiger’s Conjecture, which asserts that every graph with chromatic
number t contains a Kt minor. Norin referred to this question as the
“Dominating Hadwiger’s Conjecture” and believes it is likely false. In
this talk, we present our recent work on the Dominating Hadwiger’s
Conjecture and discuss the key ideas of our results.

Joint work with Michael Scully and Thomas Tibbetts.