Supermagic labeling of \(C_n\Box C_m\)

<p>A supermagic labeling (often also called vertex-magic edge labeling) of a graph <span class="math inline">\(G(V,E)\)</span> with <span class="math inline">\(|E|=q\)</span> is a bijection from <span class="math inline">\(E\)</span> to the set of first <span class="math inline">\(k\)</span> positive integers such that the sum of labels of all incident edges of every vertex <span class="math inline">\(x\in V\)</span> is equal to the same integer <span class="math inline">\(c\)</span>. An existence of a supermagic labeling of Cartesian product of two cycles, <span class="math inline">\(C_{n}\Box C_m\)</span> for <span class="math inline">\(n,m\geq4\)</span> and both <span class="math inline">\(n,m\)</span> even and for any <span class="math inline">\(C_n\Box C_n\)</span> with <span class="math inline">\(n\geq3\)</span> was proved by Ivančo. Ivančo also conjectured that such labeling is possible for any <span class="math inline">\(C_n\Box C_m\)</span> with <span class="math inline">\(n,m\geq3\)</span>. We prove his conjecture for all <span class="math inline">\(n,m\)</span> odd that are not relatively prime.</p>