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arXiv · 2609.35361

Disproof of the dominating Hadwiger conjecture

Freddie Illingworth·Raphael Steiner
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Abstract

Hadwiger's conjecture (1943) states that every graph $G$ with chromatic number at least $t$ contains a $K_t$-model: a collection of $t$ vertex-disjoint connected subgraphs $T_1,\dots,T_t$ such that for all $1\le i<j\le t$ some vertex in $T_j$ has a neighbour in $T_i$. Replacing "some" in this definition by "every" gives rise to the significantly stronger notion of a dominating $K_t$-model introduced by Illingworth and Wood (2024). They raised the question whether every graph of chromatic number at least $t$ contains a dominating $K_t$-model. This statement is a significant strengthening of Hadwiger's conjecture and has come to be known as the dominating Hadwiger conjecture. We provide our own exposition of a disproof of this conjecture found by ChatGPT 6 Astra Ultra. The construction is the complement of a pseudorandom triangle-free graph that is obtained by randomly subsampling a block geometric graph based on the Suzuki-Tits ovoid.

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Freddie Illingworth, Raphael Steiner. 2026-09-28. Disproof of the dominating Hadwiger conjecture. https://arxiv.org/abs/2609.35361

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