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A note on bipartite graphs whose [1,k]-domination number equal to their number of vertices [1,k]-控制数等于其顶点数的二部图的注释
arXiv: Combinatorics Pub Date : 2019-12-09 DOI: 10.7494/opmath.2020.40.3.375
N. Ghareghani, Iztok Peterin, P. Sharifani
{"title":"A note on bipartite graphs whose [1,k]-domination number equal to their number of vertices","authors":"N. Ghareghani, Iztok Peterin, P. Sharifani","doi":"10.7494/opmath.2020.40.3.375","DOIUrl":"https://doi.org/10.7494/opmath.2020.40.3.375","url":null,"abstract":"A subset $D$ of the vertex set $V$ of a graph $G$ is called an $[1,k]$-dominating set if every vertex from $V-D$ is adjacent to at least one vertex and at most $k$ vertices of $D$. A $[1,k]$-dominating set with the minimum number of vertices is called a $gamma_{[1,k]}$-set and the number of its vertices is the $[1,k]$-domination number $gamma_{[1,k]}(G)$ of $G$. In this short note we show that the decision problem whether $gamma_{[1,k]}(G)=n$ is an $NP$-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph $G$ of order $n$ satisfying $gamma_{[1,k]}(G)=n$ is given for every integer $ngeq (k+1)(2k+3)$.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79555316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Tree Containment and Degree Conditions 树的安全壳和度条件
arXiv: Combinatorics Pub Date : 2019-12-09 DOI: 10.1007/978-3-030-55857-4_19
M. Stein
{"title":"Tree Containment and Degree Conditions","authors":"M. Stein","doi":"10.1007/978-3-030-55857-4_19","DOIUrl":"https://doi.org/10.1007/978-3-030-55857-4_19","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74193995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
A combinatorial construction for two formulas in Slater’s list Slater列表中两个公式的组合构造
arXiv: Combinatorics Pub Date : 2019-12-05 DOI: 10.1142/s1793042120400114
Kagan Kursungöz
{"title":"A combinatorial construction for two formulas in Slater’s list","authors":"Kagan Kursungöz","doi":"10.1142/s1793042120400114","DOIUrl":"https://doi.org/10.1142/s1793042120400114","url":null,"abstract":"We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers-Ramanujan identities, the generating function yields two formulas in Slater's list. The same formulas were constructed by Hirschhorn. We also use staircases to give alternative triple series for partitions into $d-$distinct parts for any $d geq 2$.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79807477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes 从非对称3秩关联方案构造非交换8秩关联方案对
arXiv: Combinatorics Pub Date : 2019-11-21 DOI: 10.5802/alco.167
A. Hanaki, Masayoshi Yoshikawa
{"title":"A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes","authors":"A. Hanaki, Masayoshi Yoshikawa","doi":"10.5802/alco.167","DOIUrl":"https://doi.org/10.5802/alco.167","url":null,"abstract":"We construct a pair of non-commutative rank 8 association schemes from a rank 3 non-symmetric association scheme. For the pair, two association schemes have the same character table but different Frobenius-Schur indicators. This situation is similar to the pair of the dihedral group and the quaternion group of order 8. We also determine the structures of adjacency algebras of them over the rational number field.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73493211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
What is the Perfect Shuffle 什么是完美洗牌
arXiv: Combinatorics Pub Date : 2019-11-15 DOI: 10.1090/spec/022/01
James Enouen
{"title":"What is the Perfect Shuffle","authors":"James Enouen","doi":"10.1090/spec/022/01","DOIUrl":"https://doi.org/10.1090/spec/022/01","url":null,"abstract":"When shuffling a deck of cards, one probably wants to make sure it is thoroughly shuffled. A way to do this is by sifting through the cards to ensure that no adjacent cards are the same number, because surely this is a poorly shuffled deck. Unfortunately, human intuition for probability tends to lead us astray. For a standard 52-card deck of playing cards, the event is actually extremely likely. This report will attempt to elucidate how to answer this surprisingly difficult combinatorial question directly using rook polynomials.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83434892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Abstract 3-Rigidity and Bivariate $C_2^1$-Splines II: Combinatorial Characterization. 3-刚性和二元$C_2^1$样条II:组合表征。
arXiv: Combinatorics Pub Date : 2019-11-01 DOI: 10.19086/da.34692
K. Clinch, B. Jackson, Shin-ichi Tanigawa
{"title":"Abstract 3-Rigidity and Bivariate $C_2^1$-Splines II: Combinatorial Characterization.","authors":"K. Clinch, B. Jackson, Shin-ichi Tanigawa","doi":"10.19086/da.34692","DOIUrl":"https://doi.org/10.19086/da.34692","url":null,"abstract":"We showed in the first paper of this series that the generic $C_2^1$-cofactor matroid is the unique maximal abstract $3$-rigidity matroid. In this paper we obtain a combinatorial characterization of independence in this matroid. This solves the cofactor counterpart of the combinatorial characterization problem for the rigidity of generic 3-dimensional bar-joint frameworks. We use our characterization to verify that the counterparts of conjectures of Dress (on the rank function) and Lov'{a}sz and Yemini (which suggested a sufficient connectivity condition for rigidity) hold for this matroid.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85961551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Abstract 3-Rigidity and Bivariate $C_2^1$-Splines I: Whiteley's Maximality Conjecture 3-刚性和二元$C_2^1$样条I: Whiteley极大性猜想
arXiv: Combinatorics Pub Date : 2019-11-01 DOI: 10.19086/da.34691
K. Clinch, B. Jackson, Shin-ichi Tanigawa
{"title":"Abstract 3-Rigidity and Bivariate $C_2^1$-Splines I: Whiteley's Maximality Conjecture","authors":"K. Clinch, B. Jackson, Shin-ichi Tanigawa","doi":"10.19086/da.34691","DOIUrl":"https://doi.org/10.19086/da.34691","url":null,"abstract":"A long-standing conjecture in rigidity theory states that the generic 3-dimensional rigidity matroid is the unique maximal abstract 3-rigidity matroid (with respect to the weak order on matroids). Based on a close similarity between the generic 3-dimensional rigidity matroid and the generic $C_2^1$-cofactor matroid from approximation theory, Whiteley made an analogous conjecture in 1996 that the generic $C_2^1$-cofactor matroid is the unique maximal abstract 3-rigidity matroid. We verify Whiteley's conjecture in this paper. A key step in our proof is to verify a second conjecture of Whiteley that the `double V-replacement operation' preserves independence in the generic $C_2^1$-cofactor matroid.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85278796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Simple graphs of order 12 and minimum degree6 contain K6 minors 12阶和最小度6的简单图包含K6次图
arXiv: Combinatorics Pub Date : 2019-10-25 DOI: 10.2140/involve.2020.13.829
Ryan Odeneal, Andrei Pavelescu
{"title":"Simple graphs of order 12 and minimum degree\u00006 contain K6 minors","authors":"Ryan Odeneal, Andrei Pavelescu","doi":"10.2140/involve.2020.13.829","DOIUrl":"https://doi.org/10.2140/involve.2020.13.829","url":null,"abstract":"We prove that every simple graph of order 12 which has minimum degree 6 contains a K_6 minor.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84960230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weakly distinguishing graph polynomials on addable properties 基于可加性质的图多项式弱区分
arXiv: Combinatorics Pub Date : 2019-10-14 DOI: 10.2140/moscow.2020.9.333
J. Makowsky, Vsevolod Rakita
{"title":"Weakly distinguishing graph polynomials on addable properties","authors":"J. Makowsky, Vsevolod Rakita","doi":"10.2140/moscow.2020.9.333","DOIUrl":"https://doi.org/10.2140/moscow.2020.9.333","url":null,"abstract":"A graph polynomial $P$ is weakly distinguishing if for almost all finite graphs $G$ there is a finite graph $H$ that is not isomorphic to $G$ with $P(G)=P(H)$. It is weakly distinguishing on a graph property $mathcal{C}$ if for almost all finite graphs $Ginmathcal{C}$ there is $H in mathcal{C}$ that is not isomorphic to $G$ with $P(G)=P(H)$. We give sufficient conditions on a graph property $mathcal{C}$ for the characteristic, clique, independence, matching, and domination and $xi$ polynomials, as well as the Tutte polynomial and its specialisations, to be weakly distinguishing on $mathcal{C}$. One such condition is to be addable and small in the sense of C. McDiarmid, A. Steger and D. Welsh (2005). Another one is to be of genus at most $k$.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85283472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On the dominated chromatic number of certain graphs 论某些图的支配色数
arXiv: Combinatorics Pub Date : 2019-10-07 DOI: 10.22108/TOC.2020.119361.1675
S. Alikhani, Mohammad R. Piri
{"title":"On the dominated chromatic number of certain graphs","authors":"S. Alikhani, Mohammad R. Piri","doi":"10.22108/TOC.2020.119361.1675","DOIUrl":"https://doi.org/10.22108/TOC.2020.119361.1675","url":null,"abstract":"Let $G$ be a simple graph. The dominated coloring of $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated chromatic number of $G$, denoted by $chi_{dom}(G)$. Stability (bondage number) of dominated chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the dominated chromatic number of $G$. In this paper, we study the dominated chromatic number, dominated stability and dominated bondage number of certain graphs.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73775570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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