Rainbow subgraphs in edge-colored complete graphs: Answering two questions by Erdős and Tuza

Pub Date : 2023-12-12 DOI:10.1002/jgt.23063

Maria Axenovich, Felix C. Clemen

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Abstract

An edge-coloring of a complete graph with a set of colors $C$ is called completely balanced if any vertex is incident to the same number of edges of each color from $C$ . Erdős and Tuza asked in 1993 whether for any graph $F$ on $ℓ$ edges and any completely balanced coloring of any sufficiently large complete graph using $ℓ$ colors contains a rainbow copy of $F$ . This question was restated by Erdős in his list of “Some of my favourite problems on cycles and colourings.” We answer this question in the negative for most cliques $F = K_{q}$ by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph $F$ .

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边色完整图中的彩虹子图：回答厄尔多斯和图扎的两个问题

如果任何顶点都与 C$C$ 中每种颜色的相同数量的边相连，则具有颜色集 C$C$ 的完整图的边着色称为完全平衡着色。厄多斯和图扎在 1993 年提出了这样一个问题：对于边上有 ℓ$\ell $ 的任何图 F$F$，以及任何足够大的完整图的完全平衡着色（使用 ℓ$\ell $ 颜色）是否包含 F$F$ 的彩虹副本？厄尔多斯在他的 "我最喜欢的一些循环和着色问题 "列表中重述了这个问题。我们通过给出各自完全平衡着色的明确构造，回答了大多数小群 F=Kq$F={K}_{q}$ 的否定问题。此外，我们还回答了一个与之相关的问题，即颜色数多于图 F$F$ 中边数的完整图的完全平衡着色。

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