Properly colored and rainbow copies of graphs with few cherries

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• 作者：Benny Sudakov ^{benjamin.sudakov@math.ethz.ch} ; Jan Volec ^jan@ucw.cz
• 关键词：Ramsey-type problems ; Rainbow and properly colored subgraphs ; Spanning subgraphs ; Local lemma
• 刊名：Journal of Combinatorial Theory, Series B
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：122
• 期：Complete
• 页码：391-416
• 全文大小：539 K
• 卷排序：122

文摘

Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c   be a coloring of the edges of the complete graph KnKn such that at each vertex every color appears only constantly many times. In 1979, Shearer conjectured that such a coloring c must contain a properly colored copy of G. We establish this conjecture in a strong form, showing that it holds even for graphs G   with O(n4/3)O(n4/3) cherries and moreover this bound on the number of cherries is best possible up to a constant factor. We also prove that one can find a rainbow copy of such G   in every edge-coloring of KnKn in which all colors appear bounded number of times.Our proofs combine a framework of Lu and Székely for using the lopsided Lovász local lemma in the space of random bijections together with some additional ideas.