A study of Fibonacci cordial labeling in structured graph families

<p>A <em>Fibonacci cordial labeling</em> of a graph <span class="math inline">\(G\)</span> is an injective function <span class="math inline">\(f: V(G) \rightarrow \{F_0, F_1, \dots,\\ F_n\}\)</span>, where <span class="math inline">\(F_i\)</span> denotes the <span class="math inline">\(i^{\text{th}}\)</span> Fibonacci number, such that the induced edge labeling <span class="math inline">\(f^*: E(G) \rightarrow \{0,1\}\)</span>, given by <span class="math inline">\(f^*(uv) = (f(u) + f(v))\)</span> <span class="math inline">\((\bmod\ 2)\)</span>, satisfies the balance condition <span class="math inline">\(|e_f(0) - e_f(1)| \le 1\)</span>. Here, <span class="math inline">\(e_f(0)\)</span> and <span class="math inline">\(e_f(1)\)</span> represent the number of edges labeled 0 and 1, respectively. A graph that admits such a labeling is termed a <em>Fibonacci cordial graph</em>. In this paper, we investigate the existence and construction of Fibonacci cordial labelings for several families of graphs, including <em>Generalized Petersen graphs</em>, <em>open and closed helm graphs</em>, <em>joint sum graphs</em>, and <em>circulant graphs of small order</em>. New results and examples are presented, contributing to the growing body of knowledge on graph labelings inspired by numerical sequences.</p>