On cordial labelings of fans with other graphs. (Q2846643)

For a graph \(G\), a vertex labeling \(f\: V(G)\to \{ 0, 1\}\) induces an edge labeling \(f^{*}\: E(G)\to \{ 0, 1\}\) defined by \(f^{*}(uw)= | f(u)-f(w)| \). For \(i\in \{ 0, 1\}\), let \(v_i\) (\(e_i\)) be the number of vertices (edges), respectively, labeled with \(i\). A graph \(G\) is cordial if there exists a vertex labeling \(f\) such that \(| v_0-v_1| \leq 1\) and \(| e_0-e_1| \leq 1\). Let \(P_m\) and \(C_m\) denote a path and a cycle of order \(m\), let \(F_m\) denote a fan of order \(m+1\) and \(G_m\in \{ F_m, P_m, C_m\}\). In the paper, the authors characterize all pairs \(n\) and \(m\) for which the join \(F_n+G_m\) or the union \(F_n\cup G_m\) is a cordial graph.