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Graph classes with linear Ramsey numbers
Discrete Mathematics ( IF 0.7 ) Pub Date : 2021-01-27 , DOI: 10.1016/j.disc.2021.112307
Bogdan Alecu , Aistis Atminas , Vadim Lozin , Viktor Zamaraev

The Ramsey number RX(p,q) for a class of graphs X is the minimum n such that every graph in X with at least n vertices has either a clique of size p or an independent set of size q. We say that Ramsey numbers are linear in X if there is a constant k such that RX(p,q)k(p+q) for all p,q. In the present paper we conjecture that if X is a hereditary class defined by finitely many forbidden induced subgraphs, then Ramsey numbers are linear in X if and only if X excludes a forest, a disjoint union of cliques and their complements. We prove the “only if” part of this conjecture and verify the “if” part for a variety of classes. We also apply the notion of linearity to bipartite Ramsey numbers and reveal a number of similarities and differences between the bipartite and non-bipartite case.



中文翻译:

具有线性Ramsey数的图类

拉姆齐数 [RXpq 对于一类图 X 是最小的 ñ 这样每个图 X 至少 ñ 顶点具有大小族 p 或一组独立的尺寸 q。我们说拉姆齐数是线性X 如果有一个常数 ķ 这样 [RXpqķp+q 对全部 pq。在本文中,我们推测X 是由有限个禁止的诱导子图定义的世袭类别,则拉姆齐数在 X 当且仅当 X不包括森林,不连贯的派系及其补充。我们证明此猜想的“仅当”部分,并针对各种类别验证“如果”部分。我们还将线性概念应用于二分Ramsey数,并揭示了二分和非二分情况之间的许多相似之处和不同之处。

更新日期:2021-01-28
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