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arXiv · 2408.06746

On the locating-chromatic number of corona product of graphs

Dian Kastika Syofyan·Suhadi Wido Saputro·Edy Tri Baskoro·Ira Apni Purwasih
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Abstract

Let $G=(V,E)$ be a finite, simple, and connected graph. The locating-chromatic number of a graph $G$ can be defined as the cardinality of a minimum resolving partition of the vertex set $V(G)$ such that all vertices have different coordinates and every two adjacent vertices in $G$ is not contained in the same partition class. In this case, the coordinate of a vertex in $G$ is expressed in terms of the distances of this vertex to all partition classes. The corona product of a graph $G$ of order $n$ and a graph $H,$ denoted by $G \odot H,$ is the graph obtained by taking one copy of $G$ and $n$ copies of $H$ and joining the $i^{th}$-vertex of $G$ to every vertex in the $i^{th}$-copy of $H$. In this paper, we determine the sharp general bound of the locating-chromatic number of $G \odot H$ for $G$ is a connected graph and $H$ is an arbitrary graph, or $G$ is a tree graph and $H$ is a complement of complete graph.

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Dian Kastika Syofyan, Suhadi Wido Saputro, Edy Tri Baskoro, Ira Apni Purwasih. 2024-08-13. On the locating-chromatic number of corona product of graphs. https://arxiv.org/abs/2408.06746

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