R. M. Marcelo, M. A. Tolentino, A. Garciano, Mari-Jo P. Ruiz, Jude C. Buot
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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":0.5000,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":null,\"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. 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引用次数: 0
摘要
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]。
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.
期刊介绍:
The Journal of Interconnection Networks (JOIN) is an international scientific journal dedicated to advancing the state-of-the-art of interconnection networks. The journal addresses all aspects of interconnection networks including their theory, analysis, design, implementation and application, and corresponding issues of communication, computing and function arising from (or applied to) a variety of multifaceted networks. Interconnection problems occur at different levels in the hardware and software design of communicating entities in integrated circuits, multiprocessors, multicomputers, and communication networks as diverse as telephone systems, cable network systems, computer networks, mobile communication networks, satellite network systems, the Internet and biological systems.