A connection between locating colorings of certain join graphs with cycles in Kautz digraphs

<p>A proper <span class="math inline">\(k\)</span>-coloring <span class="math inline">\(\alpha\)</span> of a graph <span class="math inline">\(G\)</span> induces a partition <span class="math inline">\(\Pi = \{C_1, C_2, \dots, C_k\}\)</span>, where <span class="math inline">\(C_i = \{v \in V(G) \mid \alpha(v) = i\}\)</span>. The color code of a vertex <span class="math inline">\(v \in V(G)\)</span> with respect to <span class="math inline">\(\Pi\)</span> is defined as the tuple <span class="math inline">\(c_{\Pi}(v) = (d(v, C_1), d(v, C_2), \dots, d(v, C_k))\)</span>, where <span class="math inline">\(d(v, C_i)\)</span> represents the distance from <span class="math inline">\(v\)</span> to the set <span class="math inline">\(C_i\)</span>. A proper <span class="math inline">\(k\)</span>-coloring <span class="math inline">\(\alpha\)</span> is called a locating <span class="math inline">\(k\)</span>-coloring of <span class="math inline">\(G\)</span> if <span class="math inline">\(\alpha\)</span> induces a partition <span class="math inline">\(\Pi\)</span> such that for any two distinct vertices <span class="math inline">\(u, v \in V(G)\)</span>, it holds that <span class="math inline">\(c_{\Pi}(u) \neq c_{\Pi}(v)\)</span>. The locating chromatic number of <span class="math inline">\(G\)</span>, denoted <span class="math inline">\(\chi_L(G)\)</span>, is the smallest <span class="math inline">\(k\)</span> for which a locating <span class="math inline">\(k\)</span>-coloring of <span class="math inline">\(G\)</span> exists. In this paper, we establish a connection between the locating <span class="math inline">\(k\)</span>-coloring of <span class="math inline">\(C_n(1,2,\dots,t) + K_m\)</span> and the union of graphs <span class="math inline">\(\bigcup_{i=1}^p C_{n_i} + K_m\)</span>, leveraging properties of simple cycles in directed graphs. Using this connection, we determine the locating chromatic number of <span class="math inline">\(C_n(1,2,\dots,t) + K_m\)</span> for <span class="math inline">\(t = 2\)</span> and <span class="math inline">\(n \in [6, 28]\)</span>, as well as for <span class="math inline">\(t = 3\)</span> and <span class="math inline">\(n \in [8, 24]\)</span>.</p>