论文标题
每个图表都没有$ \ m natercal {k} _9^{ - 6} $ minor是$ 8 $ -Colorable
Every graph with no $\mathcal{K}_9^{-6}$ minor is $8$-colorable
论文作者
论文摘要
对于正整数$ t $和$ s $，让$ \ natercal {k} _t^{ - s} $表示从完整的图形$ k_t $获得的图形族，通过删除$ s $ edges获得。图$ g $没有$ \ mathcal {k} _t^{ - s} $ minor，如果每个$ h $ binor对于\ mathcal {k} {k} _t _t^{ - s} $没有$ h $ binor。雅各布森（Jakobsen）在著名的哈德威格（Hadwiger）的猜想中激发了1971年的雅各布森（Jakobsen）证明，每张图形没有$ \ nothcal {k} _7^{ - 2} $ minor，$ 6 $ -Colorable;最近，本作者证明，每个图形都没有$ \ MATHCAL {K} _8^{ - 4} $ MINROR是$ 7 $ -Colorable。在本文中，我们继续我们的工作，并证明没有$ \ MATHCAL {K} _9^{ - 6} $ MINROR的每个图是$ 8 $ -Colorable。我们的结果意味着，保罗·西摩（Paul Seymour）在2017年建议的$ h $ -hadwiger的猜想对于九个顶点上的所有图$ h $都是正确的，因此$ h $是$ \ mathcal {k} _9 _9 _9^{ - 6} $的每个图的子图。
For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from the complete graph $K_t$ by removing $s$ edges. A graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in \mathcal{K}_t^{-s}$. Motivated by the famous Hadwiger's Conjecture, Jakobsen in 1971 proved that every graph with no $\mathcal{K}_7^{-2}$ minor is $6$-colorable; very recently the present authors proved that every graph with no $\mathcal{K}_8^{-4}$ minor is $7$-colorable. In this paper we continue our work and prove that every graph with no $\mathcal{K}_9^{-6}$ minor is $8$-colorable. Our result implies that $H$-Hadwiger's Conjecture, suggested by Paul Seymour in 2017, is true for all graphs $H$ on nine vertices such that $H$ is a subgraph of every graph in $ \mathcal{K}_9^{-6}$.
