arXiv 2103.020342021-03-02math.CO

完全染色的插值性质失效的一致超图的存在性

The existence of uniform hypergraphs for which interpolation property of complete coloring fails

Nastaran Haghparast, Morteza Hasanvand, Yumiko Ohno

摘要：

1967年Harary，Hedetniemi和Prins证明了每个图$G$允许一个完全的$t$ -着色，对于每个具有$χ(G) \le t \le ψ(G)$的$t$，其中$χ(G)$表示$G$的色数，$ψ(G)$表示$G$的消色差数，它是$G$允许一个完全的$r$ -着色的最大数。最近，Edwards和Rz \c aesthewski（2020）证明了这个结果对于超图是不成立的，证明了对于每一个具有$k\ge 9$的整数$k$，存在一个具有完全$χ(H)$ -染色和完全$ψ(H)$ -染色的$k$ -一致超图$H$，但对于某些具有$χ(H)< t<ψ(H)$的$t$，不存在完全$t$ -染色。他们还问是否会存在这样一个例子$3$ -一致超图，并提出了另一个问题，以加强他们的结果。本文将他们的结果推广到了$k$与$k\ge 3$的所有情形，并通过给出几类$3$ -一致超图来解决他们的问题。特别地，我们反驳了Matsumoto和第三作者（2020）最近提出的一个猜想，他们提出了一个特殊的$3$ -一致超图族来满足所需的插值性质。

英文摘要：

In 1967 Harary, Hedetniemi, and Prins showed that every graph $G$ admits a complete $t$-coloring for every $t$ with $χ(G) \le t \le ψ(G)$, where $χ(G)$ denotes the chromatic number of $G$ and $ψ(G)$ denotes the achromatic number of $G$ which is the maximum number $r$ for which $G$ admits a complete $r$-coloring. Recently, Edwards and Rz\c ażewski (2020) showed that this result fails for hypergraphs by proving that for every integer $k$ with $k\ge 9$, there exists a $k$-uniform hypergraph $H$ with a complete $χ(H)$-coloring and a complete $ψ(H)$-coloring, but no complete $t$-coloring for some $t$ with $χ(H)< t<ψ(H)$. They also asked whether there would exist such an example for $3$-uniform hypergraphs and posed another problem to strengthen their result. In this paper, we generalize their result to all cases $k$ with $k\ge 3$ and settle their problems by giving several kinds of $3$-uniform hypergraphs. In particular, we disprove a recent conjecture due to Matsumoto and the third author (2020) who suggested a special family of $3$-uniform hypergraph to satisfy the desired interpolation property.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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