Pub Date : 2023-10-05DOI: 10.1142/s0129054123410083
Min Cui, Donglei Du, Ling Gai, Ruiqi Yang
Many real-world applications arising from social networks, personalized recommendations, and others, require extracting a relatively small but broadly representative portion of massive data sets. Such problems can often be formulated as maximizing a monotone set function with cardinality constraints. In this paper, we consider a streaming model where elements arrive quickly over time, and create an effective, and low-memory algorithm. First, we provide the first single-pass linear-time algorithm, which is a a deterministic algorithm, achieves an approximation ratio of [Formula: see text] for any [Formula: see text] with a query complexity of [Formula: see text] and a memory complexity of [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the submodularity ratio. However, the algorithm may produce less-than-ideal results. Our next result is to describe a multi-streaming algorithm, which is the first deterministic algorithm to attain an approximation ratio of [Formula: see text] with linear query complexity.
来自社交网络、个性化推荐等的许多实际应用程序都需要从大量数据集中提取相对较小但具有广泛代表性的部分。这类问题通常可以表述为具有基数约束的单调集函数的最大化。在本文中，我们考虑了一个元素随时间快速到达的流模型，并创建了一个有效的低内存算法。首先，我们提供了第一个单遍线性时间算法，该算法是一种确定性算法，对任意一个查询复杂度为[Formula: see text]、内存复杂度为[Formula: see text]的[Formula: see text]实现了近似比为[Formula: see text]，其中[Formula: see text]为正整数，[Formula: see text]为子模块化比。然而，该算法可能产生不太理想的结果。我们的下一个结果是描述一个多流算法，这是第一个获得线性查询复杂度近似比率的确定性算法[公式:见文本]。
{"title":"Improved Linear-Time Streaming Algorithms for Maximizing Monotone Cardinality-Constrained Set Functions","authors":"Min Cui, Donglei Du, Ling Gai, Ruiqi Yang","doi":"10.1142/s0129054123410083","DOIUrl":"https://doi.org/10.1142/s0129054123410083","url":null,"abstract":"Many real-world applications arising from social networks, personalized recommendations, and others, require extracting a relatively small but broadly representative portion of massive data sets. Such problems can often be formulated as maximizing a monotone set function with cardinality constraints. In this paper, we consider a streaming model where elements arrive quickly over time, and create an effective, and low-memory algorithm. First, we provide the first single-pass linear-time algorithm, which is a a deterministic algorithm, achieves an approximation ratio of [Formula: see text] for any [Formula: see text] with a query complexity of [Formula: see text] and a memory complexity of [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the submodularity ratio. However, the algorithm may produce less-than-ideal results. Our next result is to describe a multi-streaming algorithm, which is the first deterministic algorithm to attain an approximation ratio of [Formula: see text] with linear query complexity.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135482653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1142/s0129054123500168
Jingui Huang, Jie Chen, Yunlong Liu, Guang Xiao, Jianxin Wang
Given an [Formula: see text]-vertex graph [Formula: see text] with a fixed linear ordering of [Formula: see text] and two integers [Formula: see text], the problem fixed-order book drawing with few crossings per edge asks whether or not [Formula: see text] admits a [Formula: see text]-page book drawing where the maximum number of crossings per edge can be upper bounded by the number [Formula: see text]. This problem was posed as an open question by Bhore et al. (J. Graph Algorithms Appl. 2020). In this paper, we study this problem from the viewpoint of parameterized complexity, in particular, we develop fixed-parameter tractable algorithms for it. More specifically, we show that this problem parameterized by both the bound number [Formula: see text] of crossings per edge and the vertex cover number [Formula: see text] of the input graph admits an algorithm with running time in [Formula: see text], and this problem parameterized by both the bound number [Formula: see text] of crossings per edge and the pathwidth [Formula: see text] of the vertex ordering admits an algorithm with running time in [Formula: see text]. Our results provide a specific answer to Bhore et al.’s question.
{"title":"Parameterized Algorithms for Fixed-Order Book Drawing with Few Crossings Per Edge","authors":"Jingui Huang, Jie Chen, Yunlong Liu, Guang Xiao, Jianxin Wang","doi":"10.1142/s0129054123500168","DOIUrl":"https://doi.org/10.1142/s0129054123500168","url":null,"abstract":"Given an [Formula: see text]-vertex graph [Formula: see text] with a fixed linear ordering of [Formula: see text] and two integers [Formula: see text], the problem fixed-order book drawing with few crossings per edge asks whether or not [Formula: see text] admits a [Formula: see text]-page book drawing where the maximum number of crossings per edge can be upper bounded by the number [Formula: see text]. This problem was posed as an open question by Bhore et al. (J. Graph Algorithms Appl. 2020). In this paper, we study this problem from the viewpoint of parameterized complexity, in particular, we develop fixed-parameter tractable algorithms for it. More specifically, we show that this problem parameterized by both the bound number [Formula: see text] of crossings per edge and the vertex cover number [Formula: see text] of the input graph admits an algorithm with running time in [Formula: see text], and this problem parameterized by both the bound number [Formula: see text] of crossings per edge and the pathwidth [Formula: see text] of the vertex ordering admits an algorithm with running time in [Formula: see text]. Our results provide a specific answer to Bhore et al.’s question.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"221 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135199178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1142/s0129054123020045
Md. Saidur Rahman, Hsu-Chun Yen
{"title":"Special Issue — Graph Algorithms: Theory and Applications A Special Issue Dedicated to the Memory of Professor Takao Nishizeki: Preface","authors":"Md. Saidur Rahman, Hsu-Chun Yen","doi":"10.1142/s0129054123020045","DOIUrl":"https://doi.org/10.1142/s0129054123020045","url":null,"abstract":"","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135246550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1142/s0129054123450065
David Orellana-Martín, Luis Valencia-Cabrera, Mario J. Pérez-Jiménez
In the framework of membrane computing, several interesting results concerning frontiers of efficiency between the complexity classes P and NP have been found by using different ingredients. One of the main characteristics of cell-like membrane systems is their rooted tree-like structure, where a natural parent-children membrane relationship exists, and objects can travel through the membranes. Separation rules are used as a method to obtain an exponential workspace in terms of membranes in polynomial time. Inspired by cell meiosis, objects from the original membrane are distributed between the two new membranes. In this work, P systems with evolutional symport/antiport rules and separation rules are used to give a solution to SAT, a well known NP-complete problem. One of the advantages of this solution is the use of the environment as a passive agent.
{"title":"A Formal Verification of a SAT Solution by P Systems with Evolution Communication and Separation Rules","authors":"David Orellana-Martín, Luis Valencia-Cabrera, Mario J. Pérez-Jiménez","doi":"10.1142/s0129054123450065","DOIUrl":"https://doi.org/10.1142/s0129054123450065","url":null,"abstract":"In the framework of membrane computing, several interesting results concerning frontiers of efficiency between the complexity classes P and NP have been found by using different ingredients. One of the main characteristics of cell-like membrane systems is their rooted tree-like structure, where a natural parent-children membrane relationship exists, and objects can travel through the membranes. Separation rules are used as a method to obtain an exponential workspace in terms of membranes in polynomial time. Inspired by cell meiosis, objects from the original membrane are distributed between the two new membranes. In this work, P systems with evolutional symport/antiport rules and separation rules are used to give a solution to SAT, a well known NP-complete problem. One of the advantages of this solution is the use of the environment as a passive agent.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135199629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1142/s0129054123480064
Andreas Maletti
Generalizations of another two open conjectures, Conjectures 13 and 14, of [Lagoutte, Maletti: Survey — Weighted extended top-down tree transducers — Part III: Composition. Proc. AFCS, LNCS 7020, p. 272–308, Springer 2011] are verified. It is proved that the composition of an unambiguous weighted extended tree transducer [Formula: see text] with a nondeleting weighted top-down tree transducer [Formula: see text] can be computed by a single weighted extended tree transducer. Extending the recent confirmation of Conjecture 11 it is also proved that the composition succeeds for any weighted top-down tree transducer [Formula: see text] (even if it is not nondeleting) if the unambiguous weighted extended tree transducer [Formula: see text] is constant.
Lagoutte, Maletti的另外两个开放猜想(猜想13和猜想14)的概括:调查-加权扩展自顶向下树形传感器-第三部分:组成。Proc AFCS, LNCS 7020, p. 272-308, Springer 2011]。证明了无二义加权扩展树换能器[公式:见文]与非删除加权自顶向下树换能器[公式:见文]的组合可以由单个加权扩展树换能器计算得到。扩展了最近对推测11的证实，也证明了如果无二义加权扩展树换能器[公式:见文本]是常数，则任何加权自顶向下树换能器(即使它不是非删除的)的组合都是成功的。
{"title":"Compositions of Weighted Extended Tree Transducers — The Unambiguous Case","authors":"Andreas Maletti","doi":"10.1142/s0129054123480064","DOIUrl":"https://doi.org/10.1142/s0129054123480064","url":null,"abstract":"Generalizations of another two open conjectures, Conjectures 13 and 14, of [Lagoutte, Maletti: Survey — Weighted extended top-down tree transducers — Part III: Composition. Proc. AFCS, LNCS 7020, p. 272–308, Springer 2011] are verified. It is proved that the composition of an unambiguous weighted extended tree transducer [Formula: see text] with a nondeleting weighted top-down tree transducer [Formula: see text] can be computed by a single weighted extended tree transducer. Extending the recent confirmation of Conjecture 11 it is also proved that the composition succeeds for any weighted top-down tree transducer [Formula: see text] (even if it is not nondeleting) if the unambiguous weighted extended tree transducer [Formula: see text] is constant.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135199635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-27DOI: 10.1142/s0129054123500211
Zhenghua Xu, Lijing Zheng, Chonghui Huang
In this short note, we present two classes of (almost) perfect [Formula: see text]-nonlinear permutations over finite fields of even characteristic.
在这篇短文中，我们提出了两类(几乎)完美的[公式:见正文]——偶特征有限域上的非线性置换。
{"title":"Two New Classes of (Almost) Perfect c-Nonlinear Permutations","authors":"Zhenghua Xu, Lijing Zheng, Chonghui Huang","doi":"10.1142/s0129054123500211","DOIUrl":"https://doi.org/10.1142/s0129054123500211","url":null,"abstract":"In this short note, we present two classes of (almost) perfect [Formula: see text]-nonlinear permutations over finite fields of even characteristic.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135536217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-27DOI: 10.1142/s012905412350017x
Zhecheng Yu, Liqiong Xu, Xuemin Wu, Chuanye Zheng
Let [Formula: see text], [Formula: see text] and [Formula: see text] be non-negative integers. The generalized Johnson graph [Formula: see text] is the graph whose vertices are the [Formula: see text]-subsets of the set [Formula: see text], and two vertices are adjacent if and only if they intersect with [Formula: see text] elements. Special cases of generalized Johnson graph include the Kneser graph [Formula: see text] and the Johnson graph [Formula: see text]. These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph [Formula: see text] and [Formula: see text] by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text], and further show that the Kneser graph [Formula: see text] [Formula: see text] is super vertex-connected and super cyclically edge-connected.
{"title":"On the Super (Edge)-Connectivity of Generalized Johnson Graphs","authors":"Zhecheng Yu, Liqiong Xu, Xuemin Wu, Chuanye Zheng","doi":"10.1142/s012905412350017x","DOIUrl":"https://doi.org/10.1142/s012905412350017x","url":null,"abstract":"Let [Formula: see text], [Formula: see text] and [Formula: see text] be non-negative integers. The generalized Johnson graph [Formula: see text] is the graph whose vertices are the [Formula: see text]-subsets of the set [Formula: see text], and two vertices are adjacent if and only if they intersect with [Formula: see text] elements. Special cases of generalized Johnson graph include the Kneser graph [Formula: see text] and the Johnson graph [Formula: see text]. These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph [Formula: see text] and [Formula: see text] by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text], and further show that the Kneser graph [Formula: see text] [Formula: see text] is super vertex-connected and super cyclically edge-connected.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135535857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-25DOI: 10.1142/s0129054123460048
Qifang Su
In this paper, we consider the reconstruction of matrices whose graph is a star with root at the central vertex from given partial eigen data. Three inverse spectral problems are discussed. The necessary and sufficient conditions for the solvability of the problems are studied. Furthermore, the corresponding numerical algorithms are presented. Some numerical examples are also provided to demonstrate the applicability of the results obtained here.
{"title":"Inverse Spectral Problems for a Special Acyclic Matrix","authors":"Qifang Su","doi":"10.1142/s0129054123460048","DOIUrl":"https://doi.org/10.1142/s0129054123460048","url":null,"abstract":"In this paper, we consider the reconstruction of matrices whose graph is a star with root at the central vertex from given partial eigen data. Three inverse spectral problems are discussed. The necessary and sufficient conditions for the solvability of the problems are studied. Furthermore, the corresponding numerical algorithms are presented. Some numerical examples are also provided to demonstrate the applicability of the results obtained here.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135817219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-09DOI: 10.1142/s0129054123500193
Hui Dong
Vertex and link residual closeness are two new measures of network vulnerability due to the failure of individual vertices and links, respectively. We identify those graphs with maximum vertex/link residual closeness over all [Formula: see text]-vertex connected graphs with fixed fractional matching number.
{"title":"Vertex and Link Residual Closeness of Graphs of Given Fractional Matching Number","authors":"Hui Dong","doi":"10.1142/s0129054123500193","DOIUrl":"https://doi.org/10.1142/s0129054123500193","url":null,"abstract":"Vertex and link residual closeness are two new measures of network vulnerability due to the failure of individual vertices and links, respectively. We identify those graphs with maximum vertex/link residual closeness over all [Formula: see text]-vertex connected graphs with fixed fractional matching number.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136193151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-08DOI: 10.1142/s012905412350020x
Subhasis Koley, Sasthi C. Ghosh
For a graph [Formula: see text] and [Formula: see text], an [Formula: see text] distance coloring is a coloring [Formula: see text] of [Formula: see text] with [Formula: see text] colors such that [Formula: see text] when [Formula: see text]. Here [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] and is equal to the minimum number of edges that connect [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of [Formula: see text] distance coloring of [Formula: see text], [Formula: see text], is the minimum [Formula: see text] among all [Formula: see text] distance coloring of [Formula: see text]. A class of channel assignment problem in cellular network can be formulated as a distance graph coloring problem in regular grid graphs. The cellular network is often modelled as an infinite hexagonal grid [Formula: see text], and hence determining [Formula: see text] has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of [Formula: see text] for any odd [Formula: see text] and for even [Formula: see text], it is conjectured that [Formula: see text] where [Formula: see text] is an integer, [Formula: see text] and [Formula: see text]. For [Formula: see text], the conjecture has been proved by Ghosh and Koley [[Formula: see text]nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any even [Formula: see text].
对于一个图形[公式:见文]和[公式:见文]，一个[公式:见文]距离着色是[公式:见文]与[公式:见文]颜色的着色[公式:见文]，使得[公式:见文]在[公式:见文]时[公式:见文]。这里[Formula: see text]是[Formula: see text]和[Formula: see text]之间的距离，等于在[Formula: see text]中连接[Formula: see text]和[Formula: see text]的最小边数。[公式:见文]、[公式:见文]的[公式:见文]距离着色的跨度是所有[公式:见文]的[公式:见文]距离着色的最小[公式:见文]。蜂窝网络中的一类信道分配问题可以表述为正则网格图中的距离图着色问题。蜂窝网络通常被建模为一个无限的六边形网格[公式:见文本]，因此从实际的角度来看，确定[公式:见文本]具有相关性。Jacko和Jendrol [Discussiones Mathematicae Graph Theory, 2005]确定了对于任何奇数[公式:见文本]和偶数[公式:见文本]的确切值，推测[公式:见文本]，其中[公式:见文本]是整数，[公式:见文本]和[公式:见文本]。对于[公式:见文]，这个猜想已经被Ghosh和Koley[[公式:见文]和意大利理论计算机科学会议，2021]证明。在本文中，我们证明了任意偶数的猜想[公式:见文]。
{"title":"On the Span of ℓ Distance Coloring of Infinite Hexagonal Grid","authors":"Subhasis Koley, Sasthi C. Ghosh","doi":"10.1142/s012905412350020x","DOIUrl":"https://doi.org/10.1142/s012905412350020x","url":null,"abstract":"For a graph [Formula: see text] and [Formula: see text], an [Formula: see text] distance coloring is a coloring [Formula: see text] of [Formula: see text] with [Formula: see text] colors such that [Formula: see text] when [Formula: see text]. Here [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] and is equal to the minimum number of edges that connect [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of [Formula: see text] distance coloring of [Formula: see text], [Formula: see text], is the minimum [Formula: see text] among all [Formula: see text] distance coloring of [Formula: see text]. A class of channel assignment problem in cellular network can be formulated as a distance graph coloring problem in regular grid graphs. The cellular network is often modelled as an infinite hexagonal grid [Formula: see text], and hence determining [Formula: see text] has relevance from practical point of view. Jacko and Jendrol [Discussiones Mathematicae Graph Theory, 2005] determined the exact value of [Formula: see text] for any odd [Formula: see text] and for even [Formula: see text], it is conjectured that [Formula: see text] where [Formula: see text] is an integer, [Formula: see text] and [Formula: see text]. For [Formula: see text], the conjecture has been proved by Ghosh and Koley [[Formula: see text]nd Italian Conference on Theoretical Computer Science, 2021]. In this paper, we prove the conjecture for any even [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136362002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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