Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2022-02-10 , DOI: 10.1016/j.jctb.2022.02.002 Raphael Steiner 1
Hadwiger's conjecture states that every -minor free graph is -colorable. A qualitative strengthening of this conjecture raised by Gerards and Seymour, known as the Odd Hadwiger's conjecture, states similarly that every graph with no odd -minor is -colorable. For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on the chromatic number of the form Ct for some constant exists.
We show that if every graph without a -minor is -colorable, then every graph without an odd -minor is -colorable. As a direct corollary, we present a new state of the art bound for the chromatic number of graphs with no odd -minor. Moreover, the short proof of our result substantially simplifies the proofs of all the previous asymptotic bounds for the chromatic number of odd -minor free graphs established in a sequence of papers during the last 12 years.
中文翻译:
哈德维格猜想及其奇次变体的渐近等价
哈德维格猜想表明,每-minor 自由图是-可着色。Gerards 和 Seymour 提出的这一猜想的定性强化,称为Odd Hadwiger 猜想,类似地指出,每个没有奇数的图 -次要是-可着色。对于这两个猜想,它们的渐近松弛仍然是开放的,即,对于某个常数, Ct形式的色数是否有上限存在。
我们证明,如果每个图没有-次要是-colorable,然后每个图都没有奇数-次要是-可着色。作为直接推论,我们提出了一种新的最先进的绑定对于没有奇数的图的色数-次要的。此外,我们结果的简短证明大大简化了所有先前对奇数色数的渐近界的证明- 在过去 12 年中在一系列论文中建立的次要免费图表。