Combinatorica最新文献

{"title":"A Solution to Babai’s Problems on Digraphs with Non-diagonalizable Adjacency Matrix","authors":"Yuxuan Li, Binzhou Xia, Sanming Zhou, Wenying Zhu","doi":"10.1007/s00493-023-00068-x","DOIUrl":"https://doi.org/10.1007/s00493-023-00068-x","url":null,"abstract":"<p>The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by Babai (J Graph Theory 9:363–370, 1985). Then Babai posed the open problems of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai’s problems by constructing an infinite family of <i>s</i>-arc-transitive digraphs for each integer <span>(sge 2)</span>, and an infinite family of vertex-primitive digraphs, both of whose adjacency matrices are non-diagonalizable.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"12 10","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493575","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}

{"title":"Value Distributions of Perfect Nonlinear Functions","authors":"Lukas Kölsch, Alexandr Polujan","doi":"10.1007/s00493-023-00067-y","DOIUrl":"https://doi.org/10.1007/s00493-023-00067-y","url":null,"abstract":"<p>In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"12 11","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493574","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}

{"title":"A Generalization of the Chevalley–Warning and Ax–Katz Theorems with a View Towards Combinatorial Number Theory","authors":"David J. Grynkiewicz","doi":"10.1007/s00493-023-00057-0","DOIUrl":"https://doi.org/10.1007/s00493-023-00057-0","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;({mathbb {F}}_q)&lt;/span&gt; be a finite field of characteristic &lt;i&gt;p&lt;/i&gt; and order &lt;i&gt;q&lt;/i&gt;. The Chevalley–Warning Theorem asserts that the set &lt;i&gt;V&lt;/i&gt; of common zeros of a collection of polynomials must satisfy &lt;span&gt;(|V|equiv 0mod p)&lt;/span&gt;, provided the number of variables is sufficiently large with respect to the degrees of the polynomials. The Ax–Katz Theorem generalizes this by giving tight bounds for higher order &lt;i&gt;p&lt;/i&gt;-divisibility for |&lt;i&gt;V&lt;/i&gt;|. Besides the intrinsic algebraic interest of these results, they are also important tools in the Polynomial Method, particularly in the prime field case &lt;span&gt;({mathbb {F}}_p)&lt;/span&gt;, where they have been used to prove many results in Combinatorial Number Theory. In this paper, we begin by explaining how arguments used by Wilson to give an elementary proof of the &lt;span&gt;({mathbb {F}}_p)&lt;/span&gt; case for the Ax–Katz Theorem can also be used to prove the following generalization of the Ax–Katz Theorem for &lt;span&gt;({mathbb {F}}_p)&lt;/span&gt;, and thus also the Chevalley–Warning Theorem, where we allow varying prime power moduli. Given any box &lt;span&gt;({mathcal {B}}={mathcal {I}}_1times ldots times {mathcal {I}}_n)&lt;/span&gt;, with each &lt;span&gt;({mathcal {I}}_jsubseteq {mathbb {Z}})&lt;/span&gt; a complete system of residues modulo &lt;i&gt;p&lt;/i&gt;, and a collection of nonzero polynomials &lt;span&gt;(f_1,ldots ,f_sin {mathbb {Z}}[X_1,ldots ,X_n])&lt;/span&gt;, then the set of common zeros inside the box, &lt;/p&gt;&lt;span&gt;$$begin{aligned} V={{textbf{a}}in {mathcal {B}}:; f_1({{textbf {a}}})equiv 0mod p^{m_1},ldots ,f_s({{textbf {a}}})equiv 0mod p^{m_s}}, end{aligned}$$&lt;/span&gt;&lt;p&gt;satisfies &lt;span&gt;(|V|equiv 0mod p^m)&lt;/span&gt;, provided &lt;span&gt;(n&gt;(m-1)max _{iin [1,s]}Big {p^{m_i-1}deg f_iBig }+ sum nolimits _{i=1}^{s}frac{p^{m_i}-1}{p-1}deg f_i.)&lt;/span&gt; The introduction of the box &lt;span&gt;({mathcal {B}})&lt;/span&gt; adds a degree of flexibility, in comparison to prior work of Sun. Indeed, incorporating the ideas of Sun, a weighted version of the above result is given. We continue by explaining how the added flexibility, combined with an appropriate use of Hensel’s Lemma to choose the complete system of residues &lt;span&gt;({mathcal {I}}_j)&lt;/span&gt;, allows many combinatorial applications of the Chevalley–Warning and Ax–Katz Theorems, previously only valid for &lt;span&gt;({mathbb {F}}_p^n)&lt;/span&gt;, to extend with bare minimal modification to validity for an arbitrary finite abelian &lt;i&gt;p&lt;/i&gt;-group &lt;i&gt;G&lt;/i&gt;. We illustrate this by giving several examples, including a new proof of the exact value of the Davenport Constant &lt;span&gt;({textsf{D}}(G))&lt;/span&gt; for finite abelian &lt;i&gt;p&lt;/i&gt;-groups, and a streamlined proof of the Kemnitz Conjecture. We also derive some new results, for a finite abelian &lt;i&gt;p&lt;/i&gt;-group &lt;i&gt;G&lt;/i&gt; with exponent &lt;i&gt;q&lt;/i&gt;, regarding the constant &lt;span&gt;({textsf{s}}_{kq}(G))&lt;/span&gt;, defined as the minimal integer &lt;span&gt;(ell )&lt;/span&gt; such that any sequence of &lt;span&gt;(ell )&lt;/span&gt; terms from &lt;i&gt;G&lt;/i&gt; must contain a zero-sum subsequence of length &lt;i&gt;kq&lt;/i&gt;. ","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"12 12","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493573","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}

{"title":"Induced Subgraphs of Induced Subgraphs of Large Chromatic Number","authors":"António Girão, Freddie Illingworth, Emil Powierski, Michael Savery, Alex Scott, Youri Tamitegama, Jane Tan","doi":"10.1007/s00493-023-00061-4","DOIUrl":"https://doi.org/10.1007/s00493-023-00061-4","url":null,"abstract":"<p>We prove that, for every graph <i>F</i> with at least one edge, there is a constant <span>(c_F)</span> such that there are graphs of arbitrarily large chromatic number and the same clique number as <i>F</i> in which every <i>F</i>-free induced subgraph has chromatic number at most <span>(c_F)</span>. This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every <span>(rgeqslant 3)</span> the class of <span>(K_r)</span>-free graphs has a very strong vertex Ramsey-type property, giving a vast generalisation of a result of Folkman from 1970. We also prove related results for tournaments, hypergraphs and infinite families of graphs, and show an analogous statement for graphs where clique number is replaced by odd girth.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"12 13","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493572","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}

{"title":"Integer Multiflows in Acyclic Planar Digraphs","authors":"Guyslain Naves","doi":"10.1007/s00493-023-00065-0","DOIUrl":"https://doi.org/10.1007/s00493-023-00065-0","url":null,"abstract":"<p>We give an algorithm with complexity <span>(O((R+1)^{4k^2} k^3 n))</span> for the integer multiflow problem on instances (<i>G</i>, <i>H</i>, <i>r</i>, <i>c</i>) with <i>G</i> an acyclic planar digraph and <span>(r+c)</span> Eulerian. Here, <span>(n = |V(G)|)</span>, <span>(k = |E(H)|)</span> and <i>R</i> is the maximum request <span>(max _{h in E(H)} r(h))</span>. When <i>k</i> is fixed, this gives a polynomial-time algorithm for the arc-disjoint paths problem under the same hypothesis.Kindly check and confirm the edit made in the title.Confirmed\u0000Journal instruction requires a city and country for affiliations; however, these are missing in affiliation [1]. Please verify if the provided city is correct and amend if necessary.Since the submission, my affiliation has changed. It should now be:\u0000Laboratoire d'Informatique &amp; Systèmes, Aix-Marseille Université, CNRS UMR 7020, Marseille, France</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"13 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493242","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}

{"title":"Effective Results on the Size and Structure of Sumsets","authors":"Andrew Granville, George Shakan, Aled Walker","doi":"10.1007/s00493-023-00055-2","DOIUrl":"https://doi.org/10.1007/s00493-023-00055-2","url":null,"abstract":"<p>Let <span>(A subset {mathbb {Z}}^d)</span> be a finite set. It is known that <i>NA</i> has a particular size (<span>(vert NAvert = P_A(N))</span> for some <span>(P_A(X) in {mathbb {Q}}[X])</span>) and structure (all of the lattice points in a cone other than certain exceptional sets), once <i>N</i> is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary <i>A</i>. Such explicit results were only previously known in the special cases when <span>(d=1)</span>, when the convex hull of <i>A</i> is a simplex or when <span>(vert Avert = d+2)</span> Curran and Goldmakher (Discrete Anal. Paper No. 27, 2021), results which we improve.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"13 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493243","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}

{"title":"Polynomial Bounds for Chromatic Number. IV: A Near-polynomial Bound for Excluding the Five-vertex Path","authors":"Alex Scott, Paul Seymour, Sophie Spirkl","doi":"10.1007/s00493-023-00015-w","DOIUrl":"https://doi.org/10.1007/s00493-023-00015-w","url":null,"abstract":"<p>A graph <i>G</i> is <i>H</i><i>-free</i> if it has no induced subgraph isomorphic to <i>H</i>. We prove that a <span>(P_5)</span>-free graph with clique number <span>(omega ge 3)</span> has chromatic number at most <span>(omega ^{log _2(omega )})</span>. The best previous result was an exponential upper bound <span>((5/27)3^{omega })</span>, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds for <span>(P_5)</span>, which is the smallest open case. Thus, there is great interest in whether there is a polynomial bound for <span>(P_5)</span>-free graphs, and our result is an attempt to approach that.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"13 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493241","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}

{"title":"Separating Polynomial $$chi $$ -Boundedness from $$chi $$ -Boundedness","authors":"Marcin Briański, James Davies, Bartosz Walczak","doi":"10.1007/s00493-023-00054-3","DOIUrl":"https://doi.org/10.1007/s00493-023-00054-3","url":null,"abstract":"<p>Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function <span>(f:mathbb {N}rightarrow mathbb {N}cup {infty })</span> with <span>(f(1)=1)</span> and <span>(f(n)geqslant left( {begin{array}{c}3n+1 3end{array}}right) )</span>, we construct a hereditary class of graphs <span>({mathcal {G}})</span> such that the maximum chromatic number of a graph in <span>({mathcal {G}})</span> with clique number <i>n</i> is equal to <i>f</i>(<i>n</i>) for every <span>(nin mathbb {N})</span>. In particular, we prove that there exist hereditary classes of graphs that are <span>(chi )</span>-bounded but not polynomially <span>(chi )</span>-bounded.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"13 23","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493262","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}

{"title":"Γdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$Gamma $$end{document}-Graphic Delta-Matroids and Their Applications","authors":"Donggyu Kim, Duksang Lee, Sang-il Oum","doi":"10.1007/s00493-023-00043-6","DOIUrl":"https://doi.org/10.1007/s00493-023-00043-6","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"43 1","pages":"963 - 983"},"PeriodicalIF":1.1,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46661245","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}

{"title":"Enclosing Depth and Other Depth Measures","authors":"Patrick Schnider","doi":"10.1007/s00493-023-00045-4","DOIUrl":"https://doi.org/10.1007/s00493-023-00045-4","url":null,"abstract":"<p>We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the <i>Cascade conjecture</i>, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called <i>enclosing depth</i>, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"14 12","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493259","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}