Local Strong Rainbow Connection Number of Corona Product Between Cycle Graphs

Abstract

<p style="text-align: justify;">A rainbow geodesic is a shortest path between two vertices where all edges are colored differently. An edge coloring in which any pair of vertices with distance up to <em>d</em>, where <em>d</em> is a positive integer that can be connected by a rainbow geodesic is called <em>d</em>-local strong rainbow coloring. The <em>d</em>-local strong rainbow connection number, denoted by <em>lsrc</em><sub>d</sub>(<em>G</em>), is the least number of colors used in <em>d</em>-local strong rainbow coloring. Suppose that <em>G</em> and <em>H</em> are graphs of order <em>m</em> and <em>n</em>, respectively. The corona product of <em>G</em> and <em>H</em>, <em>G </em>⊙ <em>H</em>, is defined as a graph obtained by taking a copy of <em>G</em> and <em>m</em> copies of <em>H</em>, then connecting every vertex in the <em>i</em>-th copy of <em>H</em> to the <em>i</em>-th vertex of <em>G</em>. In this paper, we will determine the <em>lsrc</em><sub>d</sub>(<em>C</em><sub>m</sub> ⊙ <em>C</em><sub>n</sub>) for <em>d</em>=2 and <em>d</em>=3.</p>