In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $frac{1}{pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupac{s}, and many related results.
{"title":"Series acceleration formulas obtained from experimentally discovered hypergeometric recursions","authors":"P. Levrie, J. Campbell","doi":"10.46298/dmtcs.9557","DOIUrl":"https://doi.org/10.46298/dmtcs.9557","url":null,"abstract":"In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $frac{1}{pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupac{s}, and many related results.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121756718","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}
R. Ascoli, Livia Betti, J. L. Duke, Xuyan Liu, Wyatt Milgrim, Steven J. Miller, E. Palsson, F. Acosta, Santiago Velazquez Iannuzzelli
In 1946, ErdH{o}s posed the distinct distance problem, which seeks to find the minimum number of distinct distances between pairs of points selected from any configuration of $n$ points in the plane. The problem has since been explored along with many variants, including ones that extend it into higher dimensions. Less studied but no less intriguing is ErdH{o}s' distinct angle problem, which seeks to find point configurations in the plane that minimize the number of distinct angles. In their recent paper "Distinct Angles in General Position," Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum number of distinct angles in the plane in general position, which prohibits three points on any line or four on any circle. We consider the question of distinct angles in three dimensions and provide bounds on the minimum number of distinct angles in general position in this setting. We focus on pinned variants of the question, and we examine explicit constructions of point configurations in $mathbb{R}^3$ which use self-similarity to minimize the number of distinct angles. Furthermore, we study a variant of the distinct angles question regarding distinct angle chains and provide bounds on the minimum number of distinct chains in $mathbb{R}^2$ and $mathbb{R}^3$.
{"title":"Distinct Angles and Angle Chains in Three Dimensions","authors":"R. Ascoli, Livia Betti, J. L. Duke, Xuyan Liu, Wyatt Milgrim, Steven J. Miller, E. Palsson, F. Acosta, Santiago Velazquez Iannuzzelli","doi":"10.46298/dmtcs.10037","DOIUrl":"https://doi.org/10.46298/dmtcs.10037","url":null,"abstract":"In 1946, ErdH{o}s posed the distinct distance problem, which seeks to find\u0000the minimum number of distinct distances between pairs of points selected from\u0000any configuration of $n$ points in the plane. The problem has since been\u0000explored along with many variants, including ones that extend it into higher\u0000dimensions. Less studied but no less intriguing is ErdH{o}s' distinct angle\u0000problem, which seeks to find point configurations in the plane that minimize\u0000the number of distinct angles. In their recent paper \"Distinct Angles in\u0000General Position,\" Fleischmann, Konyagin, Miller, Palsson, Pesikoff, and Wolf\u0000use a logarithmic spiral to establish an upper bound of $O(n^2)$ on the minimum\u0000number of distinct angles in the plane in general position, which prohibits\u0000three points on any line or four on any circle.\u0000 We consider the question of distinct angles in three dimensions and provide\u0000bounds on the minimum number of distinct angles in general position in this\u0000setting. We focus on pinned variants of the question, and we examine explicit\u0000constructions of point configurations in $mathbb{R}^3$ which use\u0000self-similarity to minimize the number of distinct angles. Furthermore, we\u0000study a variant of the distinct angles question regarding distinct angle chains\u0000and provide bounds on the minimum number of distinct chains in $mathbb{R}^2$\u0000and $mathbb{R}^3$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116507187","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 study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds independently of the magnitude of numbers belonging to the multisets. Our heuristic can be used also for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab.
{"title":"A heuristic technique for decomposing multisets of non-negative integers according to the Minkowski sum","authors":"L. Margara","doi":"10.46298/dmtcs.9877","DOIUrl":"https://doi.org/10.46298/dmtcs.9877","url":null,"abstract":"We study the following problem. Given a multiset $M$ of non-negative\u0000integers, decide whether there exist and, in the positive case, compute two\u0000non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of\u0000two multisets A and B is a multiset containing all possible sums of any element\u0000of A and any element of B. This problem was proved to be NP-complete when\u0000multisets are replaced by sets. This version of the problem is strictly related\u0000to the factorization of boolean polynomials that turns out to be NP-complete as\u0000well. When multisets are considered, the problem is equivalent to the\u0000factorization of polynomials with non-negative integer coefficients. The\u0000computational complexity of both these problems is still unknown.\u0000 The main contribution of this paper is a heuristic technique for decomposing\u0000multisets of non-negative integers. Experimental results show that our\u0000heuristic decomposes multisets of hundreds of elements within seconds\u0000independently of the magnitude of numbers belonging to the multisets. Our\u0000heuristic can be used also for factoring polynomials in N[x]. We show that,\u0000when the degree of the polynomials gets larger, our technique is much faster\u0000than the state-of-the-art algorithms implemented in commercial software like\u0000Mathematica and MatLab.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121414536","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 mixed graph is a set of vertices together with an edge set and an arc set. An $(m,n)$-mixed graph $G$ is a mixed graph whose edges are each assigned one of $m$ colours, and whose arcs are each assigned one of $n$ colours. A emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc colours, and the arc directions of edges and arcs incident with $v$. The group of all allowed switches is $Gamma$. Let $k geq 1$ be a fixed integer and $Gamma$ a fixed permutation group. We consider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks if there a sequence of switches at vertices of $G$ with respect to $Gamma$ so that the resulting $(m,n)$-mixed graph admits a homomorphism to an $(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problem can be solved in polynomial time for $k leq 2$, and is NP-hard for $k geq 3$. This provides a step towards a general dichotomy theorem for the $Gamma$-switchable homomorphism decision problem.
{"title":"The 2-colouring problem for (m,n)-mixed graphs with switching is polynomial","authors":"R. Brewster, A. Kidner, G. MacGillivray","doi":"10.46298/dmtcs.9242","DOIUrl":"https://doi.org/10.46298/dmtcs.9242","url":null,"abstract":"A mixed graph is a set of vertices together with an edge set and an arc set.\u0000An $(m,n)$-mixed graph $G$ is a mixed graph whose edges are each assigned one\u0000of $m$ colours, and whose arcs are each assigned one of $n$ colours. A\u0000emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc\u0000colours, and the arc directions of edges and arcs incident with $v$. The group\u0000of all allowed switches is $Gamma$.\u0000 Let $k geq 1$ be a fixed integer and $Gamma$ a fixed permutation group. We\u0000consider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks if\u0000there a sequence of switches at vertices of $G$ with respect to $Gamma$ so\u0000that the resulting $(m,n)$-mixed graph admits a homomorphism to an\u0000$(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problem\u0000can be solved in polynomial time for $k leq 2$, and is NP-hard for $k geq 3$.\u0000This provides a step towards a general dichotomy theorem for the\u0000$Gamma$-switchable homomorphism decision problem.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114231632","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}
Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(geq,neq,>)$ is the same as that of $n-1-text{asc}$ on the class $I_n(>,neq,geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(geq,neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-text{asc}$ and desc on $I_n(>,neq,geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(geq,neq,>)|=|I_n(>,neq,geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.
{"title":"Further enumeration results concerning a recent equivalence of restricted inversion sequences","authors":"T. Mansour, M. Shattuck","doi":"10.46298/dmtcs.8330","DOIUrl":"https://doi.org/10.46298/dmtcs.8330","url":null,"abstract":"Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(geq,neq,>)$ is the same as that of $n-1-text{asc}$ on the class $I_n(>,neq,geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(geq,neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-text{asc}$ and desc on $I_n(>,neq,geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(geq,neq,>)|=|I_n(>,neq,geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128864679","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}
The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $pi(G)$ and the remoteness $rho(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.
{"title":"Proximity, remoteness and maximum degree in graphs","authors":"P. Dankelmann, Sonwabile Mafunda, Sufiyan Mallu","doi":"10.46298/dmtcs.9432","DOIUrl":"https://doi.org/10.46298/dmtcs.9432","url":null,"abstract":"The average distance of a vertex $v$ of a connected graph $G$ is the\u0000arithmetic mean of the distances from $v$ to all other vertices of $G$. The\u0000proximity $pi(G)$ and the remoteness $rho(G)$ of $G$ are the minimum and the\u0000maximum of the average distances of the vertices of $G$, respectively.\u0000 In this paper, we give upper bounds on the remoteness and proximity for\u0000graphs of given order, minimum degree and maximum degree. Our bounds are sharp\u0000apart from an additive constant.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116122131","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}
Marc Distel, Robert Hickingbotham, T. Huynh, D. Wood
Dujmovi'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $Gsubseteq H boxtimes P boxtimes K_{max{2g,3}}$. We improve this result by replacing "4" by "3" and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $ H boxtimes Pboxtimes K_ell$, for some planar graph $H$ with treewidth $3$, where $ell=max{2glfloor frac{d}{2} rfloor,d+3lfloorfrac{d}{2}rfloor-3}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $Hboxtimes Pboxtimes K_{max{4g,7}}$, for some planar graph $H$ with treewidth $3$.
刘建军，刘建军，刘建军，等。ACM 2020]证明了对于每个具有欧拉属的图$G$$g$，存在一个树宽接近4的图$H$和一条路径$P$，使得$Gsubseteq H boxtimes P boxtimesK_{max{2g,3}}$。我们通过将“4”替换为“3”并使用$H$ planar来改进此结果。事实上，我们用所谓的框架图证明了一个更一般的结果。这意味着每个$(g,d)$ -map图都包含在$ H boxtimesPboxtimes K_ell$中，对于树宽为$3$的平面图$H$，其中$ell=max{2glfloor frac{d}{2} rfloor,d+3lfloorfrac{d}{2}rfloor-3}$。它还意味着，对于某些具有树宽$3$的平面图形$H$，每个$(g,1)$ -平面图(即，可以在欧拉属表面$g$上绘制的图，每条边最多有一个交叉点)都包含在$Hboxtimes Pboxtimes K_{max{4g,7}}$中。
{"title":"Improved product structure for graphs on surfaces","authors":"Marc Distel, Robert Hickingbotham, T. Huynh, D. Wood","doi":"10.46298/dmtcs.8877","DOIUrl":"https://doi.org/10.46298/dmtcs.8877","url":null,"abstract":"Dujmovi'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $Gsubseteq H boxtimes P boxtimes K_{max{2g,3}}$. We improve this result by replacing \"4\" by \"3\" and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $ H boxtimes Pboxtimes K_ell$, for some planar graph $H$ with treewidth $3$, where $ell=max{2glfloor frac{d}{2} rfloor,d+3lfloorfrac{d}{2}rfloor-3}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $Hboxtimes Pboxtimes K_{max{4g,7}}$, for some planar graph $H$ with treewidth $3$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133490902","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}
The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x
{"title":"Induced betweenness in order-theoretic trees","authors":"B. Courcelle","doi":"10.46298/dmtcs.7288","DOIUrl":"https://doi.org/10.46298/dmtcs.7288","url":null,"abstract":"The ternary relation B(x,y,z) of betweenness states that an element y is between the elements x and z, in some sense depending on the considered structure. In a partially ordered set (N,≤), B(x,y,z):⇔x<y<z∨z<y<x, and the corresponding betweenness structure is (N,B). The class of betweenness structures of linear orders is first-order definable. That of partial orders is monadic second-order definable. An order-theoretic tree is a partial order such that the set of elements larger that any element is linearly ordered and any two elements have an upper-bound. Finite or infinite rooted trees ordered by the ancestor relation are order-theoretic trees. In an order-theoretic tree, B(x,y,z) means that x<y<z or z<y<x or x<y≤x⊔z or z<y≤x⊔z, where x⊔z is the least upper-bound of incomparable elements x and z. In a previous article, we established that the corresponding class of betweenness structures is monadic second-order definable.We prove here that the induced substructures of the betweenness structures of the countable order-theoretic trees form a monadic second-order definable class, denoted by IBO. The proof uses a variant of cographs, the partitioned probe cographs, and their known six finite minimal excluded induced subgraphs called the bounds of the class. This proof links two apparently unrelated topics: cographs and order-theoretic trees.However, the class IBO has finitely many bounds, i.e., minimal excluded finite induced substructures. Hence it is first-order definable. The proof of finiteness uses well-quasi-orders and does not provide the finite list of bounds. Hence, the associated first-order defining sentence is not known.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116232266","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}
Inspired by Lelek's idea from [Disjoint mappings and the span of spaces, Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span of graphs. Using this, we solve the problem of determining the emph{maximal safety distance} two players can keep at all times while traversing a graph. Moreover, their moves must be made with respect to certain move rules. For this purpose, we introduce different variants of a span of a given connected graph. All the variants model the maximum safety distance kept by two players in a graph traversal, where the players may only move with accordance to a specific set of rules, and their goal: visit either all vertices, or all edges. For each variant, we show that the solution can be obtained by considering only connected subgraphs of a graph product and the projections to the factors. We characterise graphs in which it is impossible to keep a positive safety distance at all moments in time. Finally, we present a polynomial time algorithm that determines the chosen span variant of a given graph.
{"title":"Span of a Graph: Keeping the Safety Distance","authors":"I. Banič, A. Taranenko","doi":"10.46298/dmtcs.9859","DOIUrl":"https://doi.org/10.46298/dmtcs.9859","url":null,"abstract":"Inspired by Lelek's idea from [Disjoint mappings and the span of spaces,\u0000Fund. Math. 55 (1964), 199 -- 214], we introduce the novel notion of the span\u0000of graphs. Using this, we solve the problem of determining the emph{maximal\u0000safety distance} two players can keep at all times while traversing a graph.\u0000Moreover, their moves must be made with respect to certain move rules. For this\u0000purpose, we introduce different variants of a span of a given connected graph.\u0000All the variants model the maximum safety distance kept by two players in a\u0000graph traversal, where the players may only move with accordance to a specific\u0000set of rules, and their goal: visit either all vertices, or all edges. For each\u0000variant, we show that the solution can be obtained by considering only\u0000connected subgraphs of a graph product and the projections to the factors. We\u0000characterise graphs in which it is impossible to keep a positive safety\u0000distance at all moments in time. Finally, we present a polynomial time\u0000algorithm that determines the chosen span variant of a given graph.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115145730","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}
The following problem has been known since the 80's. Let $Gamma$ be an Abelian group of order $m$ (denoted $|Gamma|=m$), and let $t$ and $m_i$, $1 leq i leq t$, be positive integers such that $sum_{i=1}^t m_i=m-1$. Determine when $Gamma^*=Gammasetminus{0}$, the set of non-zero elements of $Gamma$, can be partitioned into disjoint subsets $S_i$, $1 leq i leq t$, such that $|S_i|=m_i$ and $sum_{sin S_i}s=0$ for every $i$, $1 leq i leq t$. It is easy to check that $m_igeq 2$ (for every $i$, $1 leq i leq t$) and $|I(Gamma)|neq 1$ are necessary conditions for the existence of such partitions, where $I(Gamma)$ is the set of involutions of $Gamma$. It was proved that the condition $m_igeq 2$ is sufficient if and only if $|I(Gamma)|in{0,3}$. For other groups (i.e., for which $|I(Gamma)|neq 3$ and $|I(Gamma)|>1$), only the case of any group $Gamma$ with $Gammacong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_igeq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|Gamma|$ is large enough and $|I(Gamma)|>1$, then $m_igeq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_igeq 3$ is sufficient for $Gamma$ such that $|I(Gamma)|>1$ and $|Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.
下面这个问题从80年代就知道了。设$Gamma$为顺序为$m$(记为$|Gamma|=m$)的阿别群，设$t$、$m_i$、$1leq i leq t$为正整数，使得$sum_{i=1}^t m_i=m-1$。确定$Gamma$的非零元素集合$Gamma^*=Gammasetminus{0}$何时可以划分为不相交的子集$S_i$、$1 leq i leq t$，使得$|S_i|=m_i$、$sum_{sin S_i}s=0$对于每一个$i$、$1 leq i leqt$。很容易检查$m_igeq 2$(对于每个$i$、$1 leq i leq t$)和$|I(Gamma)|neq 1$是存在这样的分区的必要条件，其中$I(Gamma)$是$Gamma$的对合集。证明了条件$m_igeq 2$当且仅当$|I(Gamma)|in{0,3}$是充分的。对于其他组(即$|I(Gamma)|neq 3$和$|I(Gamma)|>1$)，到目前为止，只有任何组$Gamma$对于某些正整数$n$具有$Gammacong(Z_2)^n$的情况才被完全分析过，并且有几位作者独立地表明$m_igeq 3$在这种情况下是有效的。此外，最近Cichacz和Tuza证明，如果$|Gamma|$足够大，$|I(Gamma)|>1$，那么$m_igeq 4$是充分的。本文将这一结果推广到所有阶为$2^n$的阿贝尔群，即证明了条件$m_igeq 3$对于$Gamma$是充分的，使得对于每一个正整数$n$$|I(Gamma)|>1$和$|Gamma|=2^n$。我们还给出了这一结果在图示幻型和反幻型标记中的一些应用。
{"title":"Zero-sum partitions of Abelian groups of order $2^n$","authors":"Sylwia Cichacz-Przenioslo, Karol Suchan","doi":"10.46298/dmtcs.9914","DOIUrl":"https://doi.org/10.46298/dmtcs.9914","url":null,"abstract":"The following problem has been known since the 80's. Let $Gamma$ be an\u0000Abelian group of order $m$ (denoted $|Gamma|=m$), and let $t$ and $m_i$, $1\u0000leq i leq t$, be positive integers such that $sum_{i=1}^t m_i=m-1$.\u0000Determine when $Gamma^*=Gammasetminus{0}$, the set of non-zero elements of\u0000$Gamma$, can be partitioned into disjoint subsets $S_i$, $1 leq i leq t$,\u0000such that $|S_i|=m_i$ and $sum_{sin S_i}s=0$ for every $i$, $1 leq i leq\u0000t$. It is easy to check that $m_igeq 2$ (for every $i$, $1 leq i leq t$) and\u0000$|I(Gamma)|neq 1$ are necessary conditions for the existence of such\u0000partitions, where $I(Gamma)$ is the set of involutions of $Gamma$. It was\u0000proved that the condition $m_igeq 2$ is sufficient if and only if\u0000$|I(Gamma)|in{0,3}$. For other groups (i.e., for which $|I(Gamma)|neq 3$\u0000and $|I(Gamma)|>1$), only the case of any group $Gamma$ with\u0000$Gammacong(Z_2)^n$ for some positive integer $n$ has been analyzed completely\u0000so far, and it was shown independently by several authors that $m_igeq 3$ is\u0000sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if\u0000$|Gamma|$ is large enough and $|I(Gamma)|>1$, then $m_igeq 4$ is sufficient.\u0000In this paper we generalize this result for every Abelian group of order $2^n$.\u0000Namely, we show that the condition $m_igeq 3$ is sufficient for $Gamma$ such\u0000that $|I(Gamma)|>1$ and $|Gamma|=2^n$, for every positive integer $n$. We\u0000also present some applications of this result to graph magic- and\u0000anti-magic-type labelings.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130463377","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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