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Three-Cuts are a Charm: Acyclicity in 3-Connected Cubic Graphs 三切是一种魅力：三连通三次图的不环性
IF 1.1 2区 数学
Combinatorica Pub Date : 2025-02-12 DOI: 10.1007/s00493-024-00126-y
František Kardoš, Edita Máčajová, Jean Paul Zerafa
{"title":"Three-Cuts are a Charm: Acyclicity in 3-Connected Cubic Graphs","authors":"František Kardoš, Edita Máčajová, Jean Paul Zerafa","doi":"10.1007/s00493-024-00126-y","DOIUrl":"https://doi.org/10.1007/s00493-024-00126-y","url":null,"abstract":"<p>Let <i>G</i> be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the <span>(S_4)</span>-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of <i>G</i> such that the complement of their union is a bipartite subgraph of <i>G</i>. They actually show that given any <span>(1^+)</span>-factor <i>F</i> (a spanning subgraph of <i>G</i> such that its vertices have degree at least 1) and an arbitrary edge <i>e</i> of <i>G</i>, there exists a perfect matching <i>M</i> of <i>G</i> containing <i>e</i> such that <span>(Gsetminus (Fcup M))</span> is bipartite. This is a step closer to comprehend better the Fan–Raspaud Conjecture and eventually the Berge–Fulkerson Conjecture. The <span>(S_4)</span>-Conjecture, now a theorem, is also the weakest assertion in a series of three conjectures made by Mazzuoccolo in 2013, with the next stronger statement being: there exist two perfect matchings of <i>G</i> such that the complement of their union is an acyclic subgraph of <i>G</i>. Unfortunately, this conjecture is not true: Jin, Steffen, and Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts. Here we show that, despite of this, every cyclically 3-edge-connected cubic graph satisfies this second conjecture.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"16 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143393285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Constructing New Geometries: A Generalized Approach to Halving for Hypertopes 构建新几何图形：高位面减半的通用方法
IF 1.1 2区 数学
Combinatorica Pub Date : 2025-01-16 DOI: 10.1007/s00493-024-00134-y
Claudio Alexandre Piedade, Philippe Tranchida
{"title":"Constructing New Geometries: A Generalized Approach to Halving for Hypertopes","authors":"Claudio Alexandre Piedade, Philippe Tranchida","doi":"10.1007/s00493-024-00134-y","DOIUrl":"https://doi.org/10.1007/s00493-024-00134-y","url":null,"abstract":"<p>Given a residually connected incidence geometry <span>(Gamma )</span> that satisfies two conditions, denoted <span>((B_1))</span> and <span>((B_2))</span>, we construct a new geometry <span>(H(Gamma ))</span> with properties similar to those of <span>(Gamma )</span>. This new geometry <span>(H(Gamma ))</span> is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how <span>(H(Gamma ))</span> relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"95 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142986730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
How Balanced Can Permutations Be? 排列如何平衡？
IF 1.1 2区 数学
Combinatorica Pub Date : 2025-01-02 DOI: 10.1007/s00493-024-00127-x
Gal Beniamini, Nir Lavee, Nati Linial
{"title":"How Balanced Can Permutations Be?","authors":"Gal Beniamini, Nir Lavee, Nati Linial","doi":"10.1007/s00493-024-00127-x","DOIUrl":"https://doi.org/10.1007/s00493-024-00127-x","url":null,"abstract":"<p>A permutation <span>(pi in mathbb {S}_n)</span> is <i>k</i>-<i>balanced</i> if every permutation of order <i>k</i> occurs in <span>(pi )</span> equally often, through order-isomorphism. In this paper, we explicitly construct <i>k</i>-balanced permutations for <span>(k le 3)</span>, and every <i>n</i> that satisfies the necessary divisibility conditions. In contrast, we prove that for <span>(k ge 4)</span>, no such permutations exist. In fact, we show that in the case <span>(k ge 4)</span>, every <i>n</i>-element permutation is at least <span>(Omega _n(n^{k-1}))</span> far from being <i>k</i>-balanced. This lower bound is matched for <span>(k=4)</span>, by a construction based on the Erdős–Szekeres permutation.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"2 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142916857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Number of Colorings of the Middle Layers of the Hamming Cube 汉明立方体中间层的着色数
IF 1.1 2区 数学
Combinatorica Pub Date : 2025-01-02 DOI: 10.1007/s00493-024-00128-w
Lina Li, Gweneth McKinley, Jinyoung Park
{"title":"The Number of Colorings of the Middle Layers of the Hamming Cube","authors":"Lina Li, Gweneth McKinley, Jinyoung Park","doi":"10.1007/s00493-024-00128-w","DOIUrl":"https://doi.org/10.1007/s00493-024-00128-w","url":null,"abstract":"<p>For an odd integer <span>(n = 2d-1)</span>, let <span>({mathcal {B}}_d)</span> be the subgraph of the hypercube <span>(Q_n)</span> induced by the two largest layers. In this paper, we describe the typical structure of proper <i>q</i>-colorings of <span>(V({mathcal {B}}_d))</span> and give asymptotics on the number of such colorings when <i>q</i> is an even number. The proofs use various tools including information theory (entropy), Sapozhenko’s graph container method and a recently developed method of Jenssen and Perkins that combines Sapozhenko’s graph container lemma with the cluster expansion for polymer models from statistical physics.\u0000</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"24 21 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142917326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Uniacute Spherical Codes 单锐角球码
IF 1.1 2区 数学
Combinatorica Pub Date : 2025-01-02 DOI: 10.1007/s00493-024-00125-z
Saba Lepsveridze, Aleksandre Saatashvili, Yufei Zhao
{"title":"Uniacute Spherical Codes","authors":"Saba Lepsveridze, Aleksandre Saatashvili, Yufei Zhao","doi":"10.1007/s00493-024-00125-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00125-z","url":null,"abstract":"<p>A spherical <i>L</i>-code, where <span>(L subseteq [-1,infty ))</span>, consists of unit vectors in <span>(mathbb {R}^d)</span> whose pairwise inner products are contained in <i>L</i>. Determining the maximum cardinality <span>(N_L(d))</span> of an <i>L</i>-code in <span>(mathbb {R}^d)</span> is a fundamental question in discrete geometry and has been extensively investigated for various choices of <i>L</i>. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to <span>(L = {-alpha , alpha })</span>, is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that <span>(N_L(d) = O_L(d))</span> for <span>(L = [-1, -beta ] cup {alpha })</span> with <span>(alpha ,beta &gt; 0)</span> (we call such <i>L</i>-codes “uniacute”), leaving open the question of determining the leading constant factor. Balla, Dräxler, Keevash, and Sudakov proved a “uniform bound” showing <span>(limsup _{drightarrow infty } N_L(d)/d le 2p)</span> for <span>(L = [-1, -beta ] cup {alpha })</span> and <span>(p = lfloor alpha /beta rfloor + 1)</span>. For which <span>((alpha ,beta ))</span> is this uniform bound tight? We completely answer this question. We develop a framework for studying uniacute codes, including a global structure theorem showing that the Gram matrix has an approximate <i>p</i>-block structure. We also formulate a notion of “modular codes,” which we conjecture to be optimal in high dimensions.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"375 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142916856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Excluded Minors for Embeddability into a Compact Surface 紧曲面可嵌入性的排除次元
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-12-29 DOI: 10.1007/s00493-024-00129-9
Agelos Georgakopoulos
{"title":"The Excluded Minors for Embeddability into a Compact Surface","authors":"Agelos Georgakopoulos","doi":"10.1007/s00493-024-00129-9","DOIUrl":"https://doi.org/10.1007/s00493-024-00129-9","url":null,"abstract":"<p>We determine the excluded minors characterising the class of countable graphs that embed into some compact surface.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"2 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142887799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Chiral Extensions of Regular Toroids 正则环面的手性扩展
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-12-29 DOI: 10.1007/s00493-024-00132-0
Antonio Montero, Micael Toledo
{"title":"Chiral Extensions of Regular Toroids","authors":"Antonio Montero, Micael Toledo","doi":"10.1007/s00493-024-00132-0","DOIUrl":"https://doi.org/10.1007/s00493-024-00132-0","url":null,"abstract":"<p>Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational symmetry but do not admit reflections. In this paper we build chiral polytopes whose facets (maximal faces) are isomorphic to a prescribed regular cubic tessellation of the <i>n</i>-dimensional torus (<span>(n geqslant 2)</span>). As a consequence, we prove that for every <span>(d geqslant 3)</span> there exist infinitely many chiral <i>d</i>-polytopes.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"153 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142887797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A New Bound for the Fourier-Entropy-Influence Conjecture 傅里叶-熵-影响猜想的一个新界
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-12-20 DOI: 10.1007/s00493-024-00133-z
Xiao Han
{"title":"A New Bound for the Fourier-Entropy-Influence Conjecture","authors":"Xiao Han","doi":"10.1007/s00493-024-00133-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00133-z","url":null,"abstract":"<p>In this paper, we prove that the Fourier entropy of an <i>n</i>-dimensional boolean function <i>f</i> can be upper-bounded by <span>(O(I(f)+ sum limits _{kin [n]}I_k(f)log frac{1}{I_k(f)}))</span>, where <i>I</i>(<i>f</i>) is its total influence and <span>(I_k(f))</span> is the influence of the <i>k</i>-th coordinate. There is no strict quantitative relationship between our bound with the known bounds for the Fourier-Min-Entropy-Influence conjecture <span>(O(I(f)log I(f)))</span> and <span>(O(I(f)^2))</span>. The proof is elementary and uses iterative bounds on moments of Fourier coefficients over different levels to estimate the Fourier entropy as its derivative.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"69 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142939884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unfriendly Partition Conjecture Holds for Line Graphs 线形图的不友好划分猜想成立
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-12-18 DOI: 10.1007/s00493-024-00131-1
Rafał Kalinowski, Monika Pilśniak, Marcin Stawiski
{"title":"Unfriendly Partition Conjecture Holds for Line Graphs","authors":"Rafał Kalinowski, Monika Pilśniak, Marcin Stawiski","doi":"10.1007/s00493-024-00131-1","DOIUrl":"https://doi.org/10.1007/s00493-024-00131-1","url":null,"abstract":"<p>A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex <i>v</i> and every color <span>(alpha )</span>, there are at most as many edges incident to <i>v</i> colored with <span>(alpha )</span> as with all other colors. We extend some known results for finite graphs to infinite graphs, also in the list setting. In particular, we prove that every infinite graph without pendant edges has a majority edge-coloring from lists of size 4. Another interesting result states that every infinite graph without vertices of finite odd degrees admits a majority edge-coloring from lists of size 2. As a consequence of our results, we prove that line graphs of any cardinality admit majority vertex-colorings from lists of size 2, thus confirming the Unfriendly Partition Conjecture for line graphs.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"82 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Improved Upper Bound for the Size of a Trifferent Code 改进的不同代码大小的上界
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-12-18 DOI: 10.1007/s00493-024-00130-2
Siddharth Bhandari, Abhishek Khetan
{"title":"Improved Upper Bound for the Size of a Trifferent Code","authors":"Siddharth Bhandari, Abhishek Khetan","doi":"10.1007/s00493-024-00130-2","DOIUrl":"https://doi.org/10.1007/s00493-024-00130-2","url":null,"abstract":"<p>A subset <span>(mathcal {C}subseteq {0,1,2}^n)</span> is said to be a <i>trifferent</i> code (of block length <i>n</i>) if for every three distinct codewords <span>(x,y, z in mathcal {C})</span>, there is a coordinate <span>(iin {1,2,ldots ,n})</span> where they all differ, that is, <span>({x(i),y(i),z(i)})</span> is same as <span>({0,1,2})</span>. Let <i>T</i>(<i>n</i>) denote the size of the largest trifferent code of block length <i>n</i>. Understanding the asymptotic behavior of <i>T</i>(<i>n</i>) is closely related to determining the zero-error capacity of the (3/2)-channel defined by Elias (IEEE Trans Inform Theory 34(5):1070–1074, 1988), and is a long-standing open problem in the area. Elias had shown that <span>(T(n)le 2times (3/2)^n)</span> and prior to our work the best upper bound was <span>(T(n)le 0.6937 times (3/2)^n)</span> due to Kurz (Example Counterexample 5:100139, 2024). We improve this bound to <span>(T(n)le c times n^{-2/5}times (3/2)^n)</span> where <i>c</i> is an absolute constant.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"36 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142841564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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