Benjamin Hellouin de MenibusGALaC, Victor LutfallaI2M, Pascal Vanier
We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that, under some weak assumptions, this variant is undecidable regardless of the shapes, extending previous results on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set.Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that it is undecidable even in a simple setting (square shapes with small modifications).
{"title":"Decision problems on geometric tilings","authors":"Benjamin Hellouin de MenibusGALaC, Victor LutfallaI2M, Pascal Vanier","doi":"arxiv-2409.11739","DOIUrl":"https://doi.org/arxiv-2409.11739","url":null,"abstract":"We study decision problems on geometric tilings. First, we study a variant of\u0000the Domino problem where square tiles are replaced by geometric tiles of\u0000arbitrary shape. We show that, under some weak assumptions, this variant is\u0000undecidable regardless of the shapes, extending previous results on rhombus\u0000tiles. This result holds even when the geometric tiling is forced to belong to\u0000a fixed set.Second, we consider the problem of deciding whether a geometric\u0000subshift has finite local complexity, which is a common assumption when\u0000studying geometric tilings. We show that it is undecidable even in a simple\u0000setting (square shapes with small modifications).","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268821","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}
This paper introduces a new reconfiguration problem of matchings in a triangular grid graph. In this problem, we are given a nearly perfect matching in which each matching edge is labeled, and aim to transform it to a target matching by sliding edges one by one. This problem is motivated to investigate the solvability of a sliding-block puzzle called ``Gourds'' on a hexagonal grid board, introduced by Hamersma et al. [ISAAC 2020]. The main contribution of this paper is to prove that, if a triangular grid graph is factor-critical and has a vertex of degree $6$, then any two matchings can be reconfigured to each other. Moreover, for a triangular grid graph (which may not have a degree-6 vertex), we present another sufficient condition using the local connectivity. Both of our results provide broad sufficient conditions for the solvability of the Gourds puzzle on a hexagonal grid board with holes, where Hamersma et al. left it as an open question.
{"title":"Reconfiguration of labeled matchings in triangular grid graphs","authors":"Naonori Kakimura, Yuta Mishima","doi":"arxiv-2409.11723","DOIUrl":"https://doi.org/arxiv-2409.11723","url":null,"abstract":"This paper introduces a new reconfiguration problem of matchings in a\u0000triangular grid graph. In this problem, we are given a nearly perfect matching\u0000in which each matching edge is labeled, and aim to transform it to a target\u0000matching by sliding edges one by one. This problem is motivated to investigate\u0000the solvability of a sliding-block puzzle called ``Gourds'' on a hexagonal grid\u0000board, introduced by Hamersma et al. [ISAAC 2020]. The main contribution of\u0000this paper is to prove that, if a triangular grid graph is factor-critical and\u0000has a vertex of degree $6$, then any two matchings can be reconfigured to each\u0000other. Moreover, for a triangular grid graph (which may not have a degree-6\u0000vertex), we present another sufficient condition using the local connectivity.\u0000Both of our results provide broad sufficient conditions for the solvability of\u0000the Gourds puzzle on a hexagonal grid board with holes, where Hamersma et al.\u0000left it as an open question.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268820","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}
Anahí GajardoUdeC, Victor LutfallaI2M, Michaël RaoLIP
We perform intensive computations of Generalised Langton's Ants, discovering rules with a big number of highways. We depict the structure of some of them, formally proving that the number of highways which are possible for a given rule does not need to be bounded, moreover it can be infinite. The frequency of appearing of these highways is very unequal within a given generalised ant rule, in some cases these frequencies where found in a ratio of $1/10^7$ in simulations, suggesting that those highways that appears as the only possible asymptotic behaviour of some rules, might be accompanied by a big family of very infrequent ones.
{"title":"Ants on the highway","authors":"Anahí GajardoUdeC, Victor LutfallaI2M, Michaël RaoLIP","doi":"arxiv-2409.10124","DOIUrl":"https://doi.org/arxiv-2409.10124","url":null,"abstract":"We perform intensive computations of Generalised Langton's Ants, discovering\u0000rules with a big number of highways. We depict the structure of some of them,\u0000formally proving that the number of highways which are possible for a given\u0000rule does not need to be bounded, moreover it can be infinite. The frequency of\u0000appearing of these highways is very unequal within a given generalised ant\u0000rule, in some cases these frequencies where found in a ratio of $1/10^7$ in\u0000simulations, suggesting that those highways that appears as the only possible\u0000asymptotic behaviour of some rules, might be accompanied by a big family of\u0000very infrequent ones.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254771","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 present a sequential cellular automaton of radius 2 1 as a solution to the density classification task that makes use of an intermediate alphabet, and converges to a clean fixed point with no remaining auxiliary or intermediate information. We extend this solution to arbitrary finite alphabets and to configurations in higher dimensions.
{"title":"A sequential solution to the density classification task using an intermediate alphabet","authors":"Pacôme Perrotin, Pedro Paulo Balbi, Eurico Ruivo","doi":"arxiv-2409.06536","DOIUrl":"https://doi.org/arxiv-2409.06536","url":null,"abstract":"We present a sequential cellular automaton of radius 2 1 as a solution to the\u0000density classification task that makes use of an intermediate alphabet, and\u0000converges to a clean fixed point with no remaining auxiliary or intermediate\u0000information. We extend this solution to arbitrary finite alphabets and to\u0000configurations in higher dimensions.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"105 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175093","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}
Guilherme C. M. Gomes, Bruno P. Masquio, Paulo E. D. Pinto, Dieter Rautenbach, Vinicius F. dos Santos, Jayme L. Szwarcfiter, Florian Werner
A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.
{"title":"Complexity of Deciding the Equality of Matching Numbers","authors":"Guilherme C. M. Gomes, Bruno P. Masquio, Paulo E. D. Pinto, Dieter Rautenbach, Vinicius F. dos Santos, Jayme L. Szwarcfiter, Florian Werner","doi":"arxiv-2409.04855","DOIUrl":"https://doi.org/arxiv-2409.04855","url":null,"abstract":"A matching is said to be disconnected if the saturated vertices induce a\u0000disconnected subgraph and induced if the saturated vertices induce a 1-regular\u0000graph. The disconnected and induced matching numbers are defined as the maximum\u0000cardinality of such matchings, respectively, and are known to be NP-hard to\u0000compute. In this paper, we study the relationship between these two parameters\u0000and the matching number. In particular, we discuss the complexity of two\u0000decision problems; first: deciding if the matching number and disconnected\u0000matching number are equal; second: deciding if the disconnected matching number\u0000and induced matching number are equal. We show that given a bipartite graph\u0000with diameter four, deciding if the matching number and disconnected matching\u0000number are equal is NP-complete; the same holds for bipartite graphs with\u0000maximum degree three. We characterize diameter three graphs with equal matching\u0000number and disconnected matching number, which yields a polynomial time\u0000recognition algorithm. Afterwards, we show that deciding if the induced and\u0000disconnected matching numbers are equal is co-NP-complete for bipartite graphs\u0000of diameter 3. When the induced matching number is large enough compared to the\u0000maximum degree, we characterize graphs where these parameters are equal, which\u0000results in a polynomial time algorithm for bounded degree graphs.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175094","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 present a sublinear time algorithm that gives random local access to the uniform distribution over satisfying assignments to an arbitrary k-CNF formula $Phi$, at exponential clause density. Our algorithm provides memory-less query access to variable assignments, such that the output variable assignments consistently emulate a single global satisfying assignment whose law is close to the uniform distribution over satisfying assignments to $Phi$. Such models were formally defined (for the more general task of locally sampling from exponentially sized sample spaces) in 2017 by Biswas, Rubinfeld, and Yodpinyanee, who studied the analogous problem for the uniform distribution over proper q-colorings. This model extends a long line of work over multiple decades that studies sublinear time algorithms for problems in theoretical computer science. Random local access and related models have been studied for a wide variety of natural Gibbs distributions and random graphical processes. Here, we establish feasiblity of random local access models for one of the most canonical such sample spaces, the set of satisfying assignments to a k-CNF formula.
{"title":"Random local access for sampling k-SAT solutions","authors":"Dingding Dong, Nitya Mani","doi":"arxiv-2409.03951","DOIUrl":"https://doi.org/arxiv-2409.03951","url":null,"abstract":"We present a sublinear time algorithm that gives random local access to the\u0000uniform distribution over satisfying assignments to an arbitrary k-CNF formula\u0000$Phi$, at exponential clause density. Our algorithm provides memory-less query\u0000access to variable assignments, such that the output variable assignments\u0000consistently emulate a single global satisfying assignment whose law is close\u0000to the uniform distribution over satisfying assignments to $Phi$. Such models were formally defined (for the more general task of locally\u0000sampling from exponentially sized sample spaces) in 2017 by Biswas, Rubinfeld,\u0000and Yodpinyanee, who studied the analogous problem for the uniform distribution\u0000over proper q-colorings. This model extends a long line of work over multiple\u0000decades that studies sublinear time algorithms for problems in theoretical\u0000computer science. Random local access and related models have been studied for\u0000a wide variety of natural Gibbs distributions and random graphical processes.\u0000Here, we establish feasiblity of random local access models for one of the most\u0000canonical such sample spaces, the set of satisfying assignments to a k-CNF\u0000formula.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175146","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}
Simon D. Fink, Matthias Pfretzschner, Ignaz Rutter, Peter Stumpf
We consider three simple quadratic time algorithms for the problem Level Planarity and give a level-planar instance that they either falsely report as negative or for which they output a drawing that is not level planar.
{"title":"Level Planarity Is More Difficult Than We Thought","authors":"Simon D. Fink, Matthias Pfretzschner, Ignaz Rutter, Peter Stumpf","doi":"arxiv-2409.01727","DOIUrl":"https://doi.org/arxiv-2409.01727","url":null,"abstract":"We consider three simple quadratic time algorithms for the problem Level\u0000Planarity and give a level-planar instance that they either falsely report as\u0000negative or for which they output a drawing that is not level planar.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175095","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}
In this paper we study a variant of the Nuel game (a generalization of the duel) which is played in turns by $N$ players. In each turn a single player must fire at one of the other players and has a certain probability of hitting and killing his target. The players shoot in a fixed sequence and when a player is eliminated, the ``move'' passes to the next surviving player. The winner is the last surviving player. We prove that, for every $Ngeq2$, the Nuel has a stationary Nash equilibrium and provide algorithms for its computation.
{"title":"Static Nuel Games with Terminal Payoff","authors":"S. Mastrakoulis, Ath. Kehagias","doi":"arxiv-2409.01681","DOIUrl":"https://doi.org/arxiv-2409.01681","url":null,"abstract":"In this paper we study a variant of the Nuel game (a generalization of the\u0000duel) which is played in turns by $N$ players. In each turn a single player\u0000must fire at one of the other players and has a certain probability of hitting\u0000and killing his target. The players shoot in a fixed sequence and when a player\u0000is eliminated, the ``move'' passes to the next surviving player. The winner is\u0000the last surviving player. We prove that, for every $Ngeq2$, the Nuel has a\u0000stationary Nash equilibrium and provide algorithms for its computation.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"83 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175096","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}
Patrizio Angelini, Therese Biedl, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, Giuseppe Liotta, Maurizio Patrignani, Sergey Pupyrev, Ignaz Rutter, Alexander Wolff
Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that upward-$k$-planarity testing is NP-complete already for $k =1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward $1$-planar drawing in which all vertices are incident to the outer face.
并不是每个底层无向图是平面的有向无环图（DAG）都可以向上绘制平面图。我们有兴趣通过考虑 DAG 的向上 $k$ 平面图来推动向上图的概念超越平面性，在这些向上图中，边在一个共同的方向上单调递增，并且对于某个整数 $kge 1$，每条边最多交叉 $k$ 次。我们证明，对于双方外平面、立方或有界路径宽度 DAG 类，单调绘图中每条边的交叉次数一般是无界的。但是，对于外路径来说，交叉次数最多为两个，而且一般来说，交叉次数最多为带宽的二次方。从计算的角度来看，我们证明了向上的 $k$ 平面性测试在 $k =1$ 时就已经是 NP-完备的，甚至对于受限的实例，向上的平面性测试也是多项式的。从正面来看，我们可以在线性时间内判定一个单源 DAG 是否允许向上$1$-平面图，在该平面图中，所有顶点都入射到外表面。
{"title":"The Price of Upwardness","authors":"Patrizio Angelini, Therese Biedl, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, Giuseppe Liotta, Maurizio Patrignani, Sergey Pupyrev, Ignaz Rutter, Alexander Wolff","doi":"arxiv-2409.01475","DOIUrl":"https://doi.org/arxiv-2409.01475","url":null,"abstract":"Not every directed acyclic graph (DAG) whose underlying undirected graph is\u0000planar admits an upward planar drawing. We are interested in pushing the notion\u0000of upward drawings beyond planarity by considering upward $k$-planar drawings\u0000of DAGs in which the edges are monotonically increasing in a common direction\u0000and every edge is crossed at most $k$ times for some integer $k ge 1$. We show\u0000that the number of crossings per edge in a monotone drawing is in general\u0000unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth\u0000DAGs. However, it is at most two for outerpaths and it is at most quadratic in\u0000the bandwidth in general. From the computational point of view, we prove that\u0000upward-$k$-planarity testing is NP-complete already for $k =1$ and even for\u0000restricted instances for which upward planarity testing is polynomial. On the\u0000positive side, we can decide in linear time whether a single-source DAG admits\u0000an upward $1$-planar drawing in which all vertices are incident to the outer\u0000face.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175145","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}
Tapas Das, Florent Foucaud, Clara Marcille, PD Pavan, Sagnik Sen
Monitoring edge-geodetic sets in a graph are subsets of vertices such that every edge of the graph must lie on all the shortest paths between two vertices of the monitoring set. These objects were introduced in a work by Foucaud, Krishna and Ramasubramony Sulochana with relation to several prior notions in the area of network monitoring like distance edge-monitoring. In this work, we explore the extension of those notions unto oriented graphs, modelling oriented networks, and call these objects monitoring arc-geodetic sets. We also define the lower and upper monitoring arc-geodetic number of an undirected graph as the minimum and maximum of the monitoring arc-geodetic number of all orientations of the graph. We determine the monitoring arc-geodetic number of fundamental graph classes such as bipartite graphs, trees, cycles, etc. Then, we characterize the graphs for which every monitoring arc-geodetic set is the entire set of vertices, and also characterize the solutions for tournaments. We also cover some complexity aspects by studying two algorithmic problems. We show that the problem of determining if an undirected graph has an orientation with the minimal monitoring arc-geodetic set being the entire set of vertices, is NP-hard. We also show that the problem of finding a monitoring arc-geodetic set of size at most $k$ is $NP$-complete when restricted to oriented graphs with maximum degree $4$.
{"title":"Monitoring arc-geodetic sets of oriented graphs","authors":"Tapas Das, Florent Foucaud, Clara Marcille, PD Pavan, Sagnik Sen","doi":"arxiv-2409.00350","DOIUrl":"https://doi.org/arxiv-2409.00350","url":null,"abstract":"Monitoring edge-geodetic sets in a graph are subsets of vertices such that\u0000every edge of the graph must lie on all the shortest paths between two vertices\u0000of the monitoring set. These objects were introduced in a work by Foucaud,\u0000Krishna and Ramasubramony Sulochana with relation to several prior notions in\u0000the area of network monitoring like distance edge-monitoring. In this work, we explore the extension of those notions unto oriented graphs,\u0000modelling oriented networks, and call these objects monitoring arc-geodetic\u0000sets. We also define the lower and upper monitoring arc-geodetic number of an\u0000undirected graph as the minimum and maximum of the monitoring arc-geodetic\u0000number of all orientations of the graph. We determine the monitoring\u0000arc-geodetic number of fundamental graph classes such as bipartite graphs,\u0000trees, cycles, etc. Then, we characterize the graphs for which every monitoring\u0000arc-geodetic set is the entire set of vertices, and also characterize the\u0000solutions for tournaments. We also cover some complexity aspects by studying\u0000two algorithmic problems. We show that the problem of determining if an\u0000undirected graph has an orientation with the minimal monitoring arc-geodetic\u0000set being the entire set of vertices, is NP-hard. We also show that the problem\u0000of finding a monitoring arc-geodetic set of size at most $k$ is $NP$-complete\u0000when restricted to oriented graphs with maximum degree $4$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142175144","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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