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Estimating Global Subgraph Counts by Sampling 抽样估计全局子图计数
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-05-19 DOI: 10.37236/11618
S. Janson, Valentas Kurauskas
{"title":"Estimating Global Subgraph Counts by Sampling","authors":"S. Janson, Valentas Kurauskas","doi":"10.37236/11618","DOIUrl":"https://doi.org/10.37236/11618","url":null,"abstract":"We give a simple proof of a generalization of an inequality for homomorphism counts by Sidorenko (1994). A special case of our inequality says that if $d_v$ denotes the degree of a vertex $v$ in a graph $G$ and $textrm{Hom}_Delta(H,G)$ denotes the number of homomorphisms from a connected graph $H$ on $h$ vertices to $G$ which map a particular vertex of $H$ to a vertex $v$ in $G$ with $d_v ge Delta$, then $textrm{Hom}_Delta(H,G) le sum_{vin G} d_v^{h-1}mathbf{1}_{d_vge Delta}$. \u0000We use this inequality to study the minimum sample size needed to estimate the number of copies of $H$ in $G$ by sampling vertices of $G$ at random.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86267636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Generating $I$-Eigenvalue Free Threshold Graphs 生成$I$-特征值自由阈值图
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-05-19 DOI: 10.37236/11225
Luiz Emilio Allem, Elismar R. Oliveira, Fernando Tura
{"title":"Generating $I$-Eigenvalue Free Threshold Graphs","authors":"Luiz Emilio Allem, Elismar R. Oliveira, Fernando Tura","doi":"10.37236/11225","DOIUrl":"https://doi.org/10.37236/11225","url":null,"abstract":"A graph is said to be $I$-eigenvalue free if it has no eigenvalues in the interval $I$ with respect to the adjacency matrix $A$. In this paper we present twoalgorithms for generating $I$-eigenvalue free threshold graphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135626385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 Complexity of the Matroid Homomorphism Problem 矩阵同态问题的复杂性
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-05-19 DOI: 10.37236/11119
Cheolwon Heo, Hyobin Kim, Siggers Mark
{"title":"The Complexity of the Matroid Homomorphism Problem","authors":"Cheolwon Heo, Hyobin Kim, Siggers Mark","doi":"10.37236/11119","DOIUrl":"https://doi.org/10.37236/11119","url":null,"abstract":"We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $rm{Hom}_mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"86 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78187789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Tight Bound for the Number of Distinct Palindromes in a Tree 树中不同回文数目的紧界
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-04-21 DOI: 10.37236/10842
Paweł Gawrychowski, Tomasz Kociumaka, Wojciech Rytter, Tomasz Waleń
{"title":"Tight Bound for the Number of Distinct Palindromes in a Tree","authors":"Paweł Gawrychowski, Tomasz Kociumaka, Wojciech Rytter, Tomasz Waleń","doi":"10.37236/10842","DOIUrl":"https://doi.org/10.37236/10842","url":null,"abstract":"For an undirected tree with edges labeled by single letters, we consider its substrings, which are labels of the simple paths between two nodes. A palindrome is a word $w$ equal to its reverse $w^R$. We prove that the maximum number of distinct palindromic substrings in a tree of $n$ edges satisfies $text{pal}(n)=O(n^{1.5})$. This solves an open problem of Brlek, Lafrenière, and Provençal (DLT 2015), who showed that $text{pal}(n)=Omega(n^{1.5})$. Hence, we settle the tight bound of $Theta(n^{1.5})$ for the maximum palindromic complexity of trees. For standard strings, i.e., for trees that are simple paths, the maximum palindromic complexity is exactly $n+1$.
 We also propose an $O(n^{1.5} log^{0.5}{n})$-time algorithm reporting all distinct palindromes and an $O(n log^2 n)$-time algorithm finding the longest palindrome in a tree.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135463998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Spectral Radius Conditions for the Rigidity of Graphs 图刚性的谱半径条件
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-04-21 DOI: 10.37236/11308
Dandan Fan, Xueyi Huang, Huiqiu Lin
{"title":"Spectral Radius Conditions for the Rigidity of Graphs","authors":"Dandan Fan, Xueyi Huang, Huiqiu Lin","doi":"10.37236/11308","DOIUrl":"https://doi.org/10.37236/11308","url":null,"abstract":"Rigidity is the property of a structure that does not flex under an applied force. In the past several decades, the rigidity of graphs has been widely studied in discrete geometry and combinatorics. Laman (1970) obtained a combinatorial characterization of rigid graphs in $mathbb{R}^2$. Lovász and Yemini (1982) proved that every $6$-connected graph is rigid in $mathbb{R}^2$. Jackson and Jordán (2005) strengthened this result, and showed that every $6$-connected graph is globally rigid in $mathbb{R}^2$. Thus every graph with algebraic connectivity greater than $5$ is globally rigid in $mathbb{R}^2$. In 2021, Cioabă, Dewar and Gu improved this bound, and proved that every graph with minimum degree at least $6$ and algebraic connectivity greater than $2+frac{1}{delta-1}$ (resp., $2+frac{2}{delta-1}$) is rigid (resp., globally rigid) in $mathbb{R}^2$. In this paper, we study the rigidity of graphs in $mathbb{R}^2$ from the viewpoint of adjacency eigenvalues. Specifically, we provide a spectral radius condition for the rigidity (resp., globally rigidity) of $2$-connected (resp., $3$-connected) graphs with given minimum degree. Furthermore, we determine the unique graph attaining the maximum spectral radius among all minimally rigid graphs of order $n$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90893445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 2
Impartial Hypergraph Games 公正超图博弈
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-04-21 DOI: 10.37236/11665
Nándor Sieben
{"title":"Impartial Hypergraph Games","authors":"Nándor Sieben","doi":"10.37236/11665","DOIUrl":"https://doi.org/10.37236/11665","url":null,"abstract":"We study two building games and two removing games played on a finite hypergraph. In each game two players take turns selecting vertices of the hypergraph until the set of jointly selected vertices satisfies a condition related to the edges of the hypergraph. The winner is the last player able to move. The building achievement game ends as soon as the set of selected vertices contains an edge. In the building avoidance game the players are not allowed to select a set that contains an edge. The removing achievement game ends as soon as the complement of the set of selected vertices no longer contains an edge. In the removing avoidance game the players are not allowed to select a set whose complement does not contain an edge. We develop some generic tools for finding the nim-value of these games and show that the nim-value can be an arbitrary nonnegative integer. The outcome of many of these games were previously determined for several special cases in algebraic and combinatorial settings. We provide several examples and show how our tools can be used to refine these results by finding nim-values.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81032360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 1
On Enumeration and Entropy of Ribbon Tilings 关于带状切片的枚举和熵
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-04-21 DOI: 10.37236/10991
Yinsong Chen, V. Kargin
{"title":"On Enumeration and Entropy of Ribbon Tilings","authors":"Yinsong Chen, V. Kargin","doi":"10.37236/10991","DOIUrl":"https://doi.org/10.37236/10991","url":null,"abstract":"The paper considers ribbon tilings of large regions and their per-tile entropy (the logarithm of the number of tilings divided by the number of tiles). For tilings of general regions by tiles of length $n$, we give an upper bound on the per-tile entropy as $n - 1$. For growing rectangular regions,  we prove the existence of the asymptotic per tile entropy and show that it is bounded from below by $log_2 (n/e)$ and from above by $log_2(en)$. For growing generalized \"Aztec Diamond\" regions and for growing \"stair\" regions, the asymptotic per-tile entropy is calculated exactly as $1/2$ and $log_2(n + 1) - 1$, respectively.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"59 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74589673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Enumeration of Perfect Matchings of the Cartesian Products of Graphs 图的笛卡尔积的完美匹配枚举
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-04-07 DOI: 10.37236/11141
Wei Li, Yao Wang
{"title":"Enumeration of Perfect Matchings of the Cartesian Products of Graphs","authors":"Wei Li, Yao Wang","doi":"10.37236/11141","DOIUrl":"https://doi.org/10.37236/11141","url":null,"abstract":"A subgraph $ H $ of a graph $G$ is nice if $ G-V(H) $ has a perfect matching. An even cycle $ C $ in an oriented graph is oddly oriented if for either choice of direction of traversal around $ C $, the number of edges of $C$ directed along the traversal is odd. An orientation $ D $ of a graph $ G $ with an even number of vertices is Pfaffian if every nice cycle of $ G $ is oddly oriented in $ D $. Let $ P_{n} $ denote a path on $ n $ vertices. The Pfaffian graph $G times P_{2n} $ was determined by Lu and Zhang [The Pfaffian property of Cartesian products of graphs, J. Comb. Optim. 27 (2014) 530--540]. In this paper, we characterize the Pfaffian graph $ G times P_{2n+1} $ with respect to the forbidden subgraphs of $G$. We first give sufficient and necessary conditions under which $Gtimes P_{2n+1}$ ($ngeqslant 2$) is Pfaffian. Then we characterize the Pfaffian graph $ G times P_{3} $ when $G$ is a bipartite graph, and we generalize this result to the the case $G$ contains exactly one odd cycle. Following these results, we enumerate the number of perfect matchings of the Pfaffian graph $G times P_{n}$ in terms of the eigenvalues of the orientation graph of $G$, and we also count perfect matchings of some Pfaffian graph $G times P_{n}$ by the eigenvalues of $G$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"28 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81591769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Bounds on Half Graph Orders in Powers of Sparse Graphs 稀疏图幂中半图阶的界
4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-04-07 DOI: 10.37236/11063
Marek Sokołowski
{"title":"Bounds on Half Graph Orders in Powers of Sparse Graphs","authors":"Marek Sokołowski","doi":"10.37236/11063","DOIUrl":"https://doi.org/10.37236/11063","url":null,"abstract":"Half graphs and their variants, such as semi-ladders and co-matchings, are configurations that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabiański et al. (STACS; 2019) prove that in powers of sparse graphs, one cannot find arbitrarily large configurations of this kind. However, these proofs either are non-constructive, or provide only loose upper bounds on the orders of half graphs and semi-ladders.In this work we provide nearly tight asymptotic lower and upper bounds on the maximum order of half graphs, parameterized by the power, in the following classes of sparse graphs: planar graphs, graphs with bounded maximum degree, graphs with bounded pathwidth or treewidth, and graphs excluding a fixed clique as a minor.
 The most significant part of our work is the upper bound for planar graphs. Here, we employ techniques of structural graph theory to analyze semi-ladders in planar graphs via the notion of cages, which expose a topological structure in semi-ladders. As an essential building block of this proof, we also state and prove a new structural result, yielding a fully polynomial bound on the neighborhood complexity in the class of planar graphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135742548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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
Proof of a Conjecture Involving Derangements and Roots of Unity 一个涉及无序和统一根的猜想的证明
IF 0.7 4区 数学
Electronic Journal of Combinatorics Pub Date : 2023-04-07 DOI: 10.37236/11377
H. Wang, Zhi-Wei Sun
{"title":"Proof of a Conjecture Involving Derangements and Roots of Unity","authors":"H. Wang, Zhi-Wei Sun","doi":"10.37236/11377","DOIUrl":"https://doi.org/10.37236/11377","url":null,"abstract":"Let $n>1$ be an odd integer, and let $zeta$ be a primitive $n$th root of unity in the complex field. Via the Eigenvector-eigenvalue Identity, we show that$$sum_{tauin D(n-1)}mathrm{sign}(tau)prod_{j=1}^{n-1}frac{1+zeta^{j-tau(j)}}{1-zeta^{j-tau(j)}}=(-1)^{frac{n-1}{2}}frac{((n-2)!!)^2}{n},$$where $D(n-1)$ is the set of all derangements of $1,ldots,n-1$.This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $delta=0,1$ we determine the value of $det[x+m_{jk}]_{1leqslant j,kleqslant n-1}$ completely, where$$m_{jk}=begin{cases}(1+zeta^{j-k})/(1-zeta^{j-k})&text{if} jnot=k,delta&text{if} j=k.end{cases}$$","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79945746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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: 4
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