DP 颜色函数和规范标签的界限

IF 0.6 4区 数学 Q3 MATHEMATICS Graphs and Combinatorics Pub Date : 2024-05-20 DOI:10.1007/s00373-024-02794-5
Ziqing Li, Yan Yang
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引用次数: 0

摘要

DP 着色是由 Dvořák 和 Postle 引入的列表着色的一般化。让 \({\mathcal {H}}=(L,H)\) 是一个图 G 的覆盖,并且 \(P_{DP}(G,{\mathcal {H}})\) 是 G 的 \({\mathcal {H}}) 着色的个数。由考尔(Kaul)和穆德罗克(Mudrock)引入的 G 的 DP 颜色函数 \(P_{DP}(G,m)\)是 \(P_{DP}(G,{\mathcal {H}})\的最小值,最小值取自 G 的所有可能的 m 层覆盖 \({\mathcal {H}}\)。对于 n 个顶点的连通图系,我们可以根据考尔(Kaul)和穆德罗克(Mudrock)的两个结果推导出树最大化 DP 颜色函数。在本文中,我们获得了 n 个顶点 2 连通图的 DP 颜色函数的严格上限。本文的另一个关注点是覆盖中的典型标签。众所周知,如果图 G 的 m 折盖 \({\mathcal{H}}\)具有规范标签,那么 \(P_{DP}(G,{\mathcal{H}})=P(G,m)\)其中 P(G, m) 是 G 的色度多项式。我们举例说明,对于某个 m 和 G,存在一个 G 的 m 折叠盖 \({\mathcal{H}}),使得 \(P_{DP}(G,{\mathcal{H}})=P(G,m)\),但是 \({\mathcal{H}})没有规范标签。我们还证明,当 G 是单环图或 Theta 图时,对于每个 \(m\ge 3\), 如果 \(P_{DP}(G,{\mathcal {H}})=P (G,m)\), 那么 \({\mathcal {H}}) 有一个规范标签。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Bounds for DP Color Function and Canonical Labelings

The DP-coloring is a generalization of the list coloring, introduced by Dvořák and Postle. Let \({\mathcal {H}}=(L,H)\) be a cover of a graph G and \(P_{DP}(G,{\mathcal {H}})\) be the number of \({\mathcal {H}}\)-colorings of G. The DP color function \(P_{DP}(G,m)\) of G, introduced by Kaul and Mudrock, is the minimum value of \(P_{DP}(G,{\mathcal {H}})\) where the minimum is taken over all possible m-fold covers \({\mathcal {H}}\) of G. For the family of n-vertex connected graphs, one can deduce that trees maximize the DP color function, from two results of Kaul and Mudrock. In this paper we obtain tight upper bounds for the DP color function of n-vertex 2-connected graphs. Another concern in this paper is the canonical labeling in a cover. It is well known that if an m-fold cover \({\mathcal {H}}\) of a graph G has a canonical labeling, then \(P_{DP}(G,{\mathcal {H}})=P(G,m)\) in which P(Gm) is the chromatic polynomial of G. However the converse statement of this conclusion is not always true. We give examples that for some m and G, there exists an m-fold cover \({\mathcal {H}}\) of G such that \(P_{DP}(G,{\mathcal {H}})=P(G,m)\), but \({\mathcal {H}}\) has no canonical labelings. We also prove that when G is a unicyclic graph or a theta graph, for each \(m\ge 3\), if \(P_{DP}(G,{\mathcal {H}})=P (G,m)\), then \({\mathcal {H}}\) has a canonical labeling.

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来源期刊
Graphs and Combinatorics
Graphs and Combinatorics 数学-数学
CiteScore
1.00
自引率
14.30%
发文量
160
审稿时长
6 months
期刊介绍: Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers, the journal also features survey articles from authors invited by the editorial board.
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