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On minimal Ramsey graphs and Ramsey equivalence in multiple colours
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2020-03-09 , DOI: 10.1017/s0963548320000036
Dennis Clemens , Anita Liebenau , Damian Reding

For an integer q ⩾ 2, a graph G is called q-Ramsey for a graph H if every q-colouring of the edges of G contains a monochromatic copy of H. If G is q-Ramsey for H yet no proper subgraph of G has this property, then G is called q-Ramsey-minimal for H. Generalizing a statement by Burr, Nešetřil and Rödl from 1977, we prove that, for q ⩾ 3, if G is a graph that is not q-Ramsey for some graph H, then G is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H as long as H is 3-connected or isomorphic to the triangle. For such H, the following are some consequences.For 2 ⩽ r < q, every r-Ramsey-minimal graph for H is contained as an induced subgraph in an infinite number of q-Ramsey-minimal graphs for H.For every q ⩾ 3, there are q-Ramsey-minimal graphs for H of arbitrarily large maximum degree, genus and chromatic number.The collection $\{\mathcal M_q(H) \colon H \text{ is 3-connected or } K_3\}$ forms an antichain with respect to the subset relation, where $\mathcal M_q(H)$ denotes the set of all graphs that are q-Ramsey-minimal for H.We also address the question of which pairs of graphs satisfy $\mathcal M_q(H_1)=\mathcal M_q(H_2)$ , in which case H1 and H2 are called q-equivalent. We show that two graphs H1 and H2 are q-equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-equivalence. Finally we indicate that for connected graphs this implication may hold: results by Nešetřil and Rödl and by Fox, Grinshpun, Liebenau, Person and Szabó imply that the complete graph is not 2-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.

中文翻译:

关于多种颜色的最小拉姆齐图和拉姆齐等价

对于整数q⩾ 2、图G叫做q- 拉姆齐的图表H如果每个q- 边缘的着色G包含的单色副本H. 如果Gq-拉姆齐为H但没有适当的子图G有这个属性,那么G叫做q-Ramsey-最小的H. 概括 Burr、Nešetřil 和 Rödl 从 1977 年的陈述,我们证明,对于q⩾ 3,如果G是一个图,不是q- 拉姆齐的一些图表H, 然后G作为诱导子图包含在无限数量的q-Ramsey-最小图H只要H与三角形 3 连通或同构。对于这样H,以下是一些后果。2 ⩽r<q, 每一个r-Ramsey-最小图H作为诱导子图包含在无限数量的q-Ramsey-最小图H.对于每一个q⩾ 3,有q-Ramsey-最小图H最大度数、属数和色数任意大。该系列$\{\mathcal M_q(H) \colon H \text{ 是 3 连通或 } K_3\}$形成关于子集关系的反链,其中$\数学 M_q(H)$表示所有图的集合q-Ramsey-最小的H.我们还解决了哪些图对满足的问题$\mathcal M_q(H_1)=\mathcal M_q(H_2)$, 在这种情况下H1H2被称为q-相等的。我们展示了两张图H1H2q- 等价于偶数q如果它们是 2 等价的,并且一般来说q- 某些人的等价物q 3 并不一定意味着 2 等价。最后,我们指出对于连通图,这种含义可能成立:Nešetřil 和 Rödl 以及 Fox、Grinshpun、Liebenau、Person 和 Szabó 的结果暗示完整图不是 2 等价于任何其他连通图。我们证明这是任意数量颜色的情况。
更新日期:2020-03-09
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