支配哈维格猜想的两种松弛形式
Two Relaxations of the Dominating Hadwiger's Conjecture
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中文总结 AI 辅助
针对支配哈维格猜想，本文证明两种松弛形式：一是给出平均度为Ct(log t)^2的图必含支配K_t-模型的界，二是将无支配K_t-模型的图顶点划分为t-1个有界最大度的部分。
中文摘要 AI 辅助
伊林沃思（Illingworth）和伍德（Wood）近期提出了支配哈维格猜想，这是哈维格猜想的强化版本，该猜想断言：每个没有支配K_t-模型的图都是(t-1)-可着色的。我们证明了该猜想的两种松弛形式。首先，我们证明，对于某个绝对常数C，每个平均度为Ct(log t)^2的图都包含一个支配K_t-模型；该界改进了伊林沃思和伍德给出的2^{t-2}，且与最优值仅相差O(log t)因子。其次，我们证明，每个没有支配K_t-模型的图的顶点都可划分为t-1个部分，使得每个部分诱导的子图具有有界最大度。
英文摘要
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.
