具有规定汉密尔顿压缩的顶点-传递图无穷族

IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED
Klavdija Kutnar, Dragan Marušič, Andriaherimanana Sarobidy Razafimahatratra
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引用次数: 0

摘要

给定一个具有汉密尔顿循环 C 的图 X,C 的压缩系数 (\kappa (X,C)\)是 ({\textrm{Aut}}\、(C)\cap {\textrm{Aut}}\,(X)\), 而 X 的汉密尔顿压缩 \(\kappa (X)\) 是 \(\kappa (X,C)\) 的最大值,其中 C 贯穿 X 中的所有汉密尔顿循环。格雷戈尔等人(Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y,2023)提出的一个问题是,对于每一个正整数 k,是否存在无穷多个顶点-传递图(Cayley 图),其汉密尔顿压缩等于 k。由于那里给出了一个汉密尔顿压缩等于 1 的无穷个 Cayley 图族,本文在 Cayley 图的情况下,用半径积的 Cayley 图构造 \(\mathbb {Z}_p\rtimes \mathbb {Z}_k\) 完全解决了这个问题,其中 p 是素数,\(k \ge 2\) 是 \(p-1\)的除数。此外,还给出了汉密尔顿压缩等于 1 的无穷个非凯利顶点传递图系。所有这些图都是元循环图,还给出了一些关于特定阶元循环图的汉密尔顿压缩的附加结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Infinite Families of Vertex-Transitive Graphs with Prescribed Hamilton Compression

Given a graph X with a Hamilton cycle C, the compression factor \(\kappa (X,C)\) of C is the order of the largest cyclic subgroup of \({\textrm{Aut}}\,(C)\cap {\textrm{Aut}}\,(X)\), and the Hamilton compression \(\kappa (X)\) of X is the maximum of \(\kappa (X,C)\) where C runs over all Hamilton cycles in X. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor et al. (Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y, 2023) whether for every positive integer k, there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to k. Since an infinite family of Cayley graphs with Hamilton compression equal to 1 was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products \(\mathbb {Z}_p\rtimes \mathbb {Z}_k\) where p is a prime and \(k \ge 2\) a divisor of \(p-1\). Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to 1 are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given.

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来源期刊
Annals of Combinatorics
Annals of Combinatorics 数学-应用数学
CiteScore
1.00
自引率
0.00%
发文量
56
审稿时长
>12 weeks
期刊介绍: Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches
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