Pub Date : 2024-07-23DOI: 10.1142/s0219265924500142
Huan Tan, Biao Zhao
Gutman proposed a topological index called the Sombor index, which was defined as [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text] in graph [Formula: see text]. In this paper, we determine the second-minimum and second-maximum values of the Sombor index over all the unicyclic graphs of order [Formula: see text] and bicyclic graphs of order [Formula: see text].
{"title":"On Sombor Index of Unicyclic and Bicyclic Graphs","authors":"Huan Tan, Biao Zhao","doi":"10.1142/s0219265924500142","DOIUrl":"https://doi.org/10.1142/s0219265924500142","url":null,"abstract":"Gutman proposed a topological index called the Sombor index, which was defined as [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text] in graph [Formula: see text]. In this paper, we determine the second-minimum and second-maximum values of the Sombor index over all the unicyclic graphs of order [Formula: see text] and bicyclic graphs of order [Formula: see text].","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141812585","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 : 2024-05-23DOI: 10.1142/s0219265924500105
Xin Li, Wen Li, Ao Tan, Mengmeng He, Weizhen Chen
Let [Formula: see text] be a graph and [Formula: see text] be the vertex set of [Formula: see text]. If any edge appears in the unique shortest path of at least one vertex pair in set [Formula: see text], then [Formula: see text] is defined as the monitoring edge-geodetic set (MEG-set for short). We denote by [Formula: see text] the size of a smallest MEG-set of [Formula: see text]. In this paper, we studied the monitoring edge-geodetic numbers of Mycielski graph classes.
设[公式：见文本]是一个图，[公式：见文本]是[公式：见文本]的顶点集。如果任何一条边出现在集合[公式：见文本]中至少一对顶点的唯一最短路径中，那么[公式：见文本]被定义为监测边-地形集（简称 MEG-集）。我们用 [公式：见正文] 表示 [公式：见正文] 的最小 MEG 集的大小。在本文中，我们研究了 Mycielski 图类的监测边-大地数。
{"title":"Monitoring Edge-Geodetic Numbers of Mycielskian Graph Classes","authors":"Xin Li, Wen Li, Ao Tan, Mengmeng He, Weizhen Chen","doi":"10.1142/s0219265924500105","DOIUrl":"https://doi.org/10.1142/s0219265924500105","url":null,"abstract":"Let [Formula: see text] be a graph and [Formula: see text] be the vertex set of [Formula: see text]. If any edge appears in the unique shortest path of at least one vertex pair in set [Formula: see text], then [Formula: see text] is defined as the monitoring edge-geodetic set (MEG-set for short). We denote by [Formula: see text] the size of a smallest MEG-set of [Formula: see text]. In this paper, we studied the monitoring edge-geodetic numbers of Mycielski graph classes.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141103678","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 : 2024-05-10DOI: 10.1142/s0219265924500099
Pingping Li, Mingzu Zhang, Litao Guo
Embedding an interconnection network into another network is one of the main problems in parallel processing and computing systems. Interconnection networks in multiprocessing systems facilitate effective communication among various system components. Obtaining the minimum cutwidth, congestion, and wirelength in graph embedding problems is of great significance for network design and architecture, where the minimum is taken over all embedding of guest graphs into host graphs, and addressing these issues can reduce time and cost in embedded design. This work aims to accurately calculate the minimum congestion and wirelength for the [Formula: see text] (the [Formula: see text]th Cartesian product of [Formula: see text]) layout into [Formula: see text]-dimensional, [Formula: see text]-dimensional, and [Formula: see text]-dimensional grids by the optimal solution of edge isoperimetric problem on [Formula: see text].
{"title":"Minimum Congestion and Wirelength of Embedding the nth Cartesian Product of K4 into Various Kinds of Grid Networks","authors":"Pingping Li, Mingzu Zhang, Litao Guo","doi":"10.1142/s0219265924500099","DOIUrl":"https://doi.org/10.1142/s0219265924500099","url":null,"abstract":"Embedding an interconnection network into another network is one of the main problems in parallel processing and computing systems. Interconnection networks in multiprocessing systems facilitate effective communication among various system components. Obtaining the minimum cutwidth, congestion, and wirelength in graph embedding problems is of great significance for network design and architecture, where the minimum is taken over all embedding of guest graphs into host graphs, and addressing these issues can reduce time and cost in embedded design. This work aims to accurately calculate the minimum congestion and wirelength for the [Formula: see text] (the [Formula: see text]th Cartesian product of [Formula: see text]) layout into [Formula: see text]-dimensional, [Formula: see text]-dimensional, and [Formula: see text]-dimensional grids by the optimal solution of edge isoperimetric problem on [Formula: see text].","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140993360","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 : 2024-04-06DOI: 10.1142/s0219265924500063
Lan Lin, Yixun Lin
The graph bipartization problem, arising from via minimization in VLSI design and related areas, consists in finding a vertex subset [Formula: see text] of graph [Formula: see text] such that the induced subgraph [Formula: see text] is bipartite and [Formula: see text] is maximized. The problem has been proved to be NP-hard even for planar graphs and cubic graphs. On the other hand, the study of polynomial-time algorithms for typical graph classes is significant in both theoretical and applied aspects. This paper focuses on several intersection graph classes, such as line graphs, circular-arc graphs, and directed path graphs. For the line graphs, we show the NP-hardness results in general and present the polynomial-time algorithms for special cases. For circular-arc graphs and directed path graphs, we propose algorithms that improve on the previously known ones.
{"title":"Graph Bipartization and Via Minimization for Intersection Graphs","authors":"Lan Lin, Yixun Lin","doi":"10.1142/s0219265924500063","DOIUrl":"https://doi.org/10.1142/s0219265924500063","url":null,"abstract":"The graph bipartization problem, arising from via minimization in VLSI design and related areas, consists in finding a vertex subset [Formula: see text] of graph [Formula: see text] such that the induced subgraph [Formula: see text] is bipartite and [Formula: see text] is maximized. The problem has been proved to be NP-hard even for planar graphs and cubic graphs. On the other hand, the study of polynomial-time algorithms for typical graph classes is significant in both theoretical and applied aspects. This paper focuses on several intersection graph classes, such as line graphs, circular-arc graphs, and directed path graphs. For the line graphs, we show the NP-hardness results in general and present the polynomial-time algorithms for special cases. For circular-arc graphs and directed path graphs, we propose algorithms that improve on the previously known ones.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140734447","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 : 2024-04-06DOI: 10.1142/s0219265924500087
Yinfen Zhu, Xu Chen, Xing Chen
Let [Formula: see text] be a connected graph and [Formula: see text] be the adjacency matrix of [Formula: see text]. Suppose that [Formula: see text] are the eigenvalues of [Formula: see text]. In this paper, we first give a graft transformation on the spectral radius of graphs and then as their application, we determine the extremal graphs with maximum and minimum spectral radii among all clique trees. Furthermore, we also determine the unique graph with maximum spectral radius among all block graphs by using different methods.
{"title":"The Graft Transformation and Their Application on the Spectral Radius of Block Graphs","authors":"Yinfen Zhu, Xu Chen, Xing Chen","doi":"10.1142/s0219265924500087","DOIUrl":"https://doi.org/10.1142/s0219265924500087","url":null,"abstract":"Let [Formula: see text] be a connected graph and [Formula: see text] be the adjacency matrix of [Formula: see text]. Suppose that [Formula: see text] are the eigenvalues of [Formula: see text]. In this paper, we first give a graft transformation on the spectral radius of graphs and then as their application, we determine the extremal graphs with maximum and minimum spectral radii among all clique trees. Furthermore, we also determine the unique graph with maximum spectral radius among all block graphs by using different methods.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140734148","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 : 2024-04-06DOI: 10.1142/s0219265924500051
Xin Feng, Xingchao Deng, Junqing Cai
Given a graph [Formula: see text] and a positive integer [Formula: see text], the anti-van der Waerden number [Formula: see text] is defined as the minimum positive integer [Formula: see text] such that every exact [Formula: see text]-coloring of the vertices of [Formula: see text] admits a rainbow [Formula: see text]-AP. In this paper, we determine the exact value of [Formula: see text] if [Formula: see text] is a [Formula: see text]-connected outerplanar graph with diameters [Formula: see text].
{"title":"Anti-van der Waerden Numbers of Some 2-Connected Outerplanar Graphs","authors":"Xin Feng, Xingchao Deng, Junqing Cai","doi":"10.1142/s0219265924500051","DOIUrl":"https://doi.org/10.1142/s0219265924500051","url":null,"abstract":"Given a graph [Formula: see text] and a positive integer [Formula: see text], the anti-van der Waerden number [Formula: see text] is defined as the minimum positive integer [Formula: see text] such that every exact [Formula: see text]-coloring of the vertices of [Formula: see text] admits a rainbow [Formula: see text]-AP. In this paper, we determine the exact value of [Formula: see text] if [Formula: see text] is a [Formula: see text]-connected outerplanar graph with diameters [Formula: see text].","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140735102","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 : 2024-03-15DOI: 10.1142/s021926592450004x
Timmy Tomy Thalavayalil, Johan Kok, S. Naduvath
The exact deg-centric graph of a simple, connected graph [Formula: see text], denoted by [Formula: see text], is a graph constructed from [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, the concepts of exact deg-centric graphs and iterated exact deg-centrication of a graph are introduced and discussed.
{"title":"A Study on Exact Deg-Centric Graphs of Graphs","authors":"Timmy Tomy Thalavayalil, Johan Kok, S. Naduvath","doi":"10.1142/s021926592450004x","DOIUrl":"https://doi.org/10.1142/s021926592450004x","url":null,"abstract":"The exact deg-centric graph of a simple, connected graph [Formula: see text], denoted by [Formula: see text], is a graph constructed from [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, the concepts of exact deg-centric graphs and iterated exact deg-centrication of a graph are introduced and discussed.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140239910","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 : 2024-03-07DOI: 10.1142/s0219265924500038
Kaiyue Meng, Yuxing Yang
Graph embedding is a fundamental problem in computer science. Let [Formula: see text] (resp., [Formula: see text]) be the interconnection network for a parallel computer system [Formula: see text] (resp., [Formula: see text]). If [Formula: see text] could be embedded into [Formula: see text], then [Formula: see text] can simulate [Formula: see text]’s behavior. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is a node-symmetric and link-symmetric recursive interconnection network for parallel computer systems. Let [Formula: see text] be a prescribed linear forest of [Formula: see text], and let [Formula: see text] and [Formula: see text] be any two distinct nodes in [Formula: see text] such that [Formula: see text] has no path with [Formula: see text] or [Formula: see text] as internal nodes, or both as end-nodes. This paper shows that there is a Hamiltonian path passing through [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] with [Formula: see text] and odd [Formula: see text] even if the number of links in [Formula: see text] is up to [Formula: see text].
{"title":"Embedding Hamiltonian Paths with Prescribed Linear Forests into k-ary n-Cube Networks","authors":"Kaiyue Meng, Yuxing Yang","doi":"10.1142/s0219265924500038","DOIUrl":"https://doi.org/10.1142/s0219265924500038","url":null,"abstract":"Graph embedding is a fundamental problem in computer science. Let [Formula: see text] (resp., [Formula: see text]) be the interconnection network for a parallel computer system [Formula: see text] (resp., [Formula: see text]). If [Formula: see text] could be embedded into [Formula: see text], then [Formula: see text] can simulate [Formula: see text]’s behavior. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is a node-symmetric and link-symmetric recursive interconnection network for parallel computer systems. Let [Formula: see text] be a prescribed linear forest of [Formula: see text], and let [Formula: see text] and [Formula: see text] be any two distinct nodes in [Formula: see text] such that [Formula: see text] has no path with [Formula: see text] or [Formula: see text] as internal nodes, or both as end-nodes. This paper shows that there is a Hamiltonian path passing through [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] with [Formula: see text] and odd [Formula: see text] even if the number of links in [Formula: see text] is up to [Formula: see text].","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140077673","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 : 2024-02-28DOI: 10.1142/s0219265924500014
Yingying Zhang, Fanfan Wang, Chenxu Yang
Motivated by the problem of network monitoring, Foucaud, Krishna and Ramasubramony Sulochana introduced the concept of monitoring edge-geodetic set and a related graph invariant. A monitoring edge-geodetic set (MEG-set for short) is a set such that the removal of any edge changes the distance between some pair of vertices in the set. The minimum size of the monitoring edge-geodetic set is called the monitoring edge-geodetic number. In this paper, we study the monitoring edge-geodetic numbers of some well-known networks, including folded hypercube, folded [Formula: see text]-cube, prism graph, anti-prism graph, Jahangir graph and windmill graphs.
受网络监控问题的启发，Foucaud、Krishna 和 Ramasubramony Sulochana 提出了监控边-大地集的概念和相关的图不变式。监控边-大地集（简称 MEG 集）是这样一个集合：移除任何一条边都会改变集合中某对顶点之间的距离。监测边-大地集的最小大小称为监测边-大地数。本文研究了一些著名网络的监测边几何数，包括折叠超立方图、折叠[公式：见正文]立方图、棱柱图、反棱柱图、贾汉吉尔图和风车图。
{"title":"Monitoring Edge-Geodetic Numbers of Some Networks","authors":"Yingying Zhang, Fanfan Wang, Chenxu Yang","doi":"10.1142/s0219265924500014","DOIUrl":"https://doi.org/10.1142/s0219265924500014","url":null,"abstract":"Motivated by the problem of network monitoring, Foucaud, Krishna and Ramasubramony Sulochana introduced the concept of monitoring edge-geodetic set and a related graph invariant. A monitoring edge-geodetic set (MEG-set for short) is a set such that the removal of any edge changes the distance between some pair of vertices in the set. The minimum size of the monitoring edge-geodetic set is called the monitoring edge-geodetic number. In this paper, we study the monitoring edge-geodetic numbers of some well-known networks, including folded hypercube, folded [Formula: see text]-cube, prism graph, anti-prism graph, Jahangir graph and windmill graphs.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140417833","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 : 2024-02-28DOI: 10.1142/s0219265924500026
R. M. Marcelo, M. A. Tolentino, A. Garciano, Mari-Jo P. Ruiz, Jude C. Buot
Let [Formula: see text] be a red–white coloring of the vertices of a nontrivial connected graph [Formula: see text] with diameter [Formula: see text], where at least one vertex is colored red. Then [Formula: see text] is called an identification coloring or simply, an ID-coloring, if and only if for any two vertices [Formula: see text] and [Formula: see text], [Formula: see text], where for any vertex [Formula: see text], [Formula: see text] and [Formula: see text] is the number of red vertices at a distance [Formula: see text] from [Formula: see text]. A graph is said to be an ID-graph if it possesses an ID-coloring. If [Formula: see text] is an ID-graph, then the spectrum of [Formula: see text] is the set of all positive integers [Formula: see text] for which [Formula: see text] has an ID-coloring with [Formula: see text] red vertices. The identification number or ID-number of a graph is the smallest element in its spectrum. In this paper, we extend a result of Kono and Zhang on the identification number of grids [Formula: see text]. In particular, we give a formulation of strong ID-coloring and use it to give a sufficient condition for an ID-coloring of a graph to be extendable to an ID-coloring of the Cartesian product of a path [Formula: see text] with [Formula: see text]. Consequently, some elements of the spectrum of grids [Formula: see text] for positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], are obtained. The complete spectrum of ladders [Formula: see text] is then determined using systematic constructions of ID-colorings of the ladders.
设[公式：见正文]是一个直径为[公式：见正文]的非三维连通图[公式：见正文]顶点的红白着色，其中至少有一个顶点被染成红色。当且仅当对于任意两个顶点[式：见文字]和[式：见文字]，[式：见文字]，其中对于任意顶点[式：见文字]，[式：见文字]和[式：见文字]是距离[式：见文字][式：见文字]的红色顶点的个数时，[式：见文字]称为标识着色或简称 ID 着色。如果一个图具有 ID 着色，则称其为 ID 图。如果[公式：见正文]是一个 ID 图，那么[公式：见正文]的谱就是所有正整数[公式：见正文]的集合，对于这些正整数[公式：见正文]，[公式：见正文]具有[公式：见正文]红色顶点的 ID 染色。图的标识号或 ID 号是图谱中最小的元素。在本文中，我们扩展了 Kono 和 Zhang 关于网格标识号的一个结果 [公式：见正文]。特别是，我们给出了强 ID 染色的表述，并用它给出了一个充分条件，即图的 ID 染色可以扩展为路径[公式：见正文]与[公式：见正文]的笛卡尔乘积的 ID 染色。因此，得到了正整数[公式：见正文]和[公式：见正文]与[公式：见正文]的网格谱[公式：见正文]的一些元素。然后，利用梯形 ID 着色的系统构造，就可以确定完整的梯形谱[式：见正文]。
{"title":"On the Vertex Identification Spectra of Grids","authors":"R. M. Marcelo, M. A. Tolentino, A. Garciano, Mari-Jo P. Ruiz, Jude C. Buot","doi":"10.1142/s0219265924500026","DOIUrl":"https://doi.org/10.1142/s0219265924500026","url":null,"abstract":"Let [Formula: see text] be a red–white coloring of the vertices of a nontrivial connected graph [Formula: see text] with diameter [Formula: see text], where at least one vertex is colored red. Then [Formula: see text] is called an identification coloring or simply, an ID-coloring, if and only if for any two vertices [Formula: see text] and [Formula: see text], [Formula: see text], where for any vertex [Formula: see text], [Formula: see text] and [Formula: see text] is the number of red vertices at a distance [Formula: see text] from [Formula: see text]. A graph is said to be an ID-graph if it possesses an ID-coloring. If [Formula: see text] is an ID-graph, then the spectrum of [Formula: see text] is the set of all positive integers [Formula: see text] for which [Formula: see text] has an ID-coloring with [Formula: see text] red vertices. The identification number or ID-number of a graph is the smallest element in its spectrum. In this paper, we extend a result of Kono and Zhang on the identification number of grids [Formula: see text]. In particular, we give a formulation of strong ID-coloring and use it to give a sufficient condition for an ID-coloring of a graph to be extendable to an ID-coloring of the Cartesian product of a path [Formula: see text] with [Formula: see text]. Consequently, some elements of the spectrum of grids [Formula: see text] for positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], are obtained. The complete spectrum of ladders [Formula: see text] is then determined using systematic constructions of ID-colorings of the ladders.","PeriodicalId":53990,"journal":{"name":"JOURNAL OF INTERCONNECTION NETWORKS","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140422122","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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