Pub Date : 2019-12-09DOI: 10.7494/opmath.2020.40.3.375
N. Ghareghani, Iztok Peterin, P. Sharifani
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 $ngeq (k+1)(2k+3)$.
{"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":"128 1","pages":""},"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}
Pub Date : 2019-12-05DOI: 10.1142/s1793042120400114
Kagan Kursungöz
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 $d geq 2$.
{"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":"1 1","pages":""},"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}
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.
{"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":"119 1","pages":""},"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}
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.
{"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":"8 1","pages":""},"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}
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.
{"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":"37 1","pages":""},"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}
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.
{"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":"16 1","pages":""},"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}
Pub Date : 2019-10-25DOI: 10.2140/involve.2020.13.829
Ryan Odeneal, Andrei Pavelescu
We prove that every simple graph of order 12 which has minimum degree 6 contains a K_6 minor.
我们证明了每一个最小度为6的12阶简单图都包含一个K_6次图。
{"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":"68 1","pages":""},"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}
Pub Date : 2019-10-14DOI: 10.2140/moscow.2020.9.333
J. Makowsky, Vsevolod Rakita
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 $k$.
对于几乎所有有限图$G$，一个图多项式$P$是弱区分是否有一个有限图$H$不同构于$G$且$P(G)=P(H)$。对于几乎所有的有限图$Ginmathcal{C}$存在$H inmathcal{C}$不同构于$G$且$P(G)=P(H)$，这是弱区分图性质$mathcal{C}$的。给出了图性质$mathcal{C}$上特征、团、独立、匹配、支配和$xi$多项式以及Tutte多项式及其专门化在$mathcal{C}$上弱区分的充分条件。其中一个条件是C. McDiarmid, A. Steger和D. Welsh(2005)意义上的可添加和小。另一种是最多有$k$属。
{"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":"50 1","pages":""},"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}
Pub Date : 2019-10-07DOI: 10.22108/TOC.2020.119361.1675
S. Alikhani, Mohammad R. Piri
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.
{"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":"40 1","pages":""},"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}
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