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arXiv · 2608.12126

Two Relaxations of the Dominating Hadwiger's Conjecture

António Girão·Sergey Norin·Youri Tamitegama·Jane Tan
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Abstract

Illingworth and Wood recently proposed the Dominating Hadwiger's Conjecture, a strengthening of Hadwiger's Conjecture which asserts that every graph with no dominating $K_t$-model is $(t-1)$-colorable. We prove two relaxations of this conjecture. First, we show that every graph with average degree $Ct (\log t)^2$ contains a dominating $K_t$-model for some absolute constant $C$. This bound improves on the $2^{t-2}$ due to Illingworth and Wood and is within an $O(\log t)$ factor from optimal. Second, we prove that the vertices of every graph with no dominating $K_t$-model can be partitioned into $t-1$ parts such that the subgraph induced by each part has bounded maximum degree.

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António Girão, Sergey Norin, Youri Tamitegama, Jane Tan. 2026-08-12. Two Relaxations of the Dominating Hadwiger's Conjecture. https://arxiv.org/abs/2608.12126

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