Extremal Results for Graphs Avoiding a Rainbow Subgraph

We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edg...

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Bibliographic Details
Published inThe Electronic journal of combinatorics Vol. 31; no. 1
Main Authors He, Zhen, Frankl, Peter, Győri, Ervin, Lv, Zequn, Salia, Nika, Tompkins, Casey, Varga, Kitti, Zhu, Xiutao
Format Journal Article
LanguageEnglish
Published 26.01.2024
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Summary:We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding a rainbow path of length three.
ISSN:1077-8926
1077-8926
DOI:10.37236/11676