完全多部图的非同构(简单)k-角双嵌入的个数

IF 0.6 3区 数学 Q3 MATHEMATICS
Simone Costa, Anita Pasotti
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引用次数: 4

摘要

本文旨在给出完全多部图的非同构k-多边形面-2可着色嵌入(有时被称为双嵌入,滥用符号)到可定向曲面的数量的指数下界。为此,我们使用了Archdeacon在2015年提出的Heffer数组及其与图嵌入的关系的概念。特别地,我们表明,在某些假设下,从单个Heffter数组中,我们可以获得指数数量的不同图嵌入。从Cavenagh, Donovan和Yazıcı在2020年构建的数组开始利用这一思想,我们得到,对于无限多个k和v值,Kv至少存在kk/2 + o(k)⋅2v⋅H(1/4)/(2k)^2 + o(v)个非同构的k-多边形面2色嵌入,其中H(⋅)为二进制熵。此外,对于Kv/t × t的嵌入,对于t∈{1,2,k},我们给出了一个2v·H(1/4)/2k(k−1)+ o(v,k)非同构的k-多边形面-2色嵌入的构造,当k为奇数且v属于宽无穷一族时。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the number of non-isomorphic (simple) k-gonal biembeddings of complete multipartite graphs
This article aims to provide exponential lower bounds on the number of non-isomorphic k-gonal face-2-colourable embeddings (sometimes called, with abuse of notation, biembeddings) of the complete multipartite graph into orientable surfaces. For this purpose, we use the concept, introduced by Archdeacon in 2015, of Heffer array and its relations with graph embeddings. In particular we show that, under certain hypotheses, from a single Heffter array, we can obtain an exponential number of distinct graph embeddings. Exploiting this idea starting from the arrays constructed by Cavenagh, Donovan and Yazıcı in 2020, we obtain that, for infinitely many values of k and v, there are at least kk/2 + o(k) ⋅ 2v ⋅ H(1/4)/(2k)^2 + o(v) non-isomorphic k-gonal face-2-colourable embeddings of Kv, where H(⋅) is the binary entropy. Moreover about the embeddings of Kv/t × t, for t ∈ {1, 2, k}, we provide a construction of 2v ⋅ H(1/4)/2k(k−1) + o(v,k) non-isomorphic k-gonal face-2-colourable embeddings whenever k is odd and v belongs to a wide infinite family of values.
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来源期刊
Ars Mathematica Contemporanea
Ars Mathematica Contemporanea MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.70
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: Ars mathematica contemporanea will publish high-quality articles in contemporary mathematics that arise from the discrete and concrete mathematics paradigm. It will favor themes that combine at least two different fields of mathematics. In particular, we welcome papers intersecting discrete mathematics with other branches of mathematics, such as algebra, geometry, topology, theoretical computer science, and combinatorics. The name of the journal was chosen carefully. Symmetry is certainly a theme that is quite welcome to the journal, as it is through symmetry that mathematics comes closest to art.
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