{"title":"The Hamilton Compression of Highly Symmetric Graphs","authors":"Petr Gregor, Arturo Merino, Torsten Mütze","doi":"10.1007/s00026-023-00674-y","DOIUrl":null,"url":null,"abstract":"<div><p>We say that a Hamilton cycle <span>\\(C=(x_1,\\ldots ,x_n)\\)</span> in a graph <i>G</i> is <i>k</i>-symmetric, if the mapping <span>\\(x_i\\mapsto x_{i+n/k}\\)</span> for all <span>\\(i=1,\\ldots ,n\\)</span>, where indices are considered modulo <i>n</i>, is an automorphism of <i>G</i>. In other words, if we lay out the vertices <span>\\(x_1,\\ldots ,x_n\\)</span> equidistantly on a circle and draw the edges of <i>G</i> as straight lines, then the drawing of <i>G</i> has <i>k</i>-fold rotational symmetry, i.e., all information about the graph is compressed into a <span>\\(360^\\circ /k\\)</span> wedge of the drawing. The maximum <i>k</i> for which there exists a <i>k</i>-symmetric Hamilton cycle in <i>G</i> is referred to as the <i>Hamilton compression of</i> <i>G</i>. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00674-y.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00674-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We say that a Hamilton cycle \(C=(x_1,\ldots ,x_n)\) in a graph G is k-symmetric, if the mapping \(x_i\mapsto x_{i+n/k}\) for all \(i=1,\ldots ,n\), where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices \(x_1,\ldots ,x_n\) equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a \(360^\circ /k\) wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression ofG. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.
如果对于所有 \(i=1,\ldots,n\)(这里的索引都是以 n 为模的),映射 \(x_i\mapstox_{i+n/k}\)是 G 的自动变形,那么我们就说图 G 中的汉密尔顿循环 \(C=(x_1,\ldots ,x_n)\)是 k 对称的。换句话说,如果我们把顶点 \(x_1,\ldots ,x_n\) 等距地画在一个圆上,并把 G 的边画成直线,那么 G 的画法就具有 k 倍旋转对称性,也就是说、关于图形的所有信息都被压缩到了绘图的一个 (360^\circ /k\)楔形中。我们研究了四个不同顶点变换图族的汉密尔顿压缩,它们分别是超立方体图、约翰逊图、永恒面图和无性群的卡莱图。在几种情况下,我们精确地确定了它们的汉密尔顿压缩率,在其他情况下,我们提供了接近的下限和上限。所构建的循环比文献中已知的几种经典格雷码的压缩率要高得多。我们的构造还产生了位串、组合和排列的灰色代码,这些代码的轨迹很少,而且/或者是平衡的。
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