论文标题
每张图无$ \ Mathcal {k} _8^{ - 4} $ MINROR是$ 7 $ -Colorable
Every graph with no $\mathcal{K}_8^{-4}$ minor is $7$-colorable
论文作者
论文摘要
哈德威格（Hadwiger）的猜想1943年指出，每张无$ k_ {t} $ binor的图形为$（t-1）$ - 可着色；对于所有$ t \ ge 7 $，它仍然很开放。对于正整数$ 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。我们说，如果每个$ h $ binor对于每个$ h $ binor，则每个$ h $ binor对于\ mathcal {k} {k} _t _t^{ - s} $，则图$ g $没有$ \ mathcal {k} _t^{ - s} $ minor。雅各布森（Jakobsen）在1971年证明，每个图都没有$ \ mathcal {k} _7^{ - 2} $ minor是$ 6 $ -Colorable。在本文中，我们考虑下一步，并证明每个图都没有$ \ MATHCAL {K} _8^{ - 4} $ MINROR是$ 7 $ -Colorable。我们的结果意味着，保罗·西摩（Paul Seymour）在2017年提出的$ h $ - 哈德威格（Paul Seymour）的猜想是八个顶点上的每张图$ h $ to的，以至于$ h $的补充具有最高学位至少四个，一个完美的匹配，一个三角形，三角形和一个长度为四个。我们的证明利用了$ \ Mathcal {k} _8^{ - 4} $未成年人在本文中获得的极端功能，角色和第二作者的广泛性kempe链条链接链条链，以及从K_5 $ k_5 $ k_5 $ graphss by Kawarabayashi and toft;这种方法是由罗伯逊（Robertson），西摩（Seymour）和托马斯（Thomas）于1993年首次开发的，以证明哈德威格（Hadwiger）的猜想是$ t = 6 $。
Hadwiger's Conjecture from 1943 states that every graph with no $K_{t}$ minor is $(t-1)$-colorable; it remains wide open for all $t\ge 7$. 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. We say that a graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in \mathcal{K}_t^{-s}$. Jakobsen in 1971 proved that every graph with no $\mathcal{K}_7^{-2}$ minor is $6$-colorable. In this paper we consider the next step and prove that every graph with no $\mathcal{K}_8^{-4}$ minor is $7$-colorable. Our result implies that $H$-Hadwiger's Conjecture, suggested by Paul Seymour in 2017, is true for every graph $H$ on eight vertices such that the complement of $H$ has maximum degree at least four, a perfect matching, a triangle and a cycle of length four. Our proof utilizes an extremal function for $\mathcal{K}_8^{-4}$ minors obtained in this paper, generalized Kempe chains of contraction-critical graphs by Rolek and the second author, and the method for finding $K_7$ minors from three different $K_5$ subgraphs by Kawarabayashi and Toft; this method was first developed by Robertson, Seymour and Thomas in 1993 to prove Hadwiger's Conjecture for $t=6$.
