On super edge-magic deficiency of unicyclic graphs (Q2811837)

A simple, finite, and undirected graph \(G(V, E)\) is edge-magic if, for some bijective function \(\phi: V \cup E \mapsto \{1, 2, \ldots, |V|+|E|\}\), \(\phi(x)+\phi(xy)+\phi(y)\) is a constant for each and every edge \(xy \in E\). Furthermore, such a graph is super edge-magic if \(\phi(V)=\{1, 2, \ldots, |V|\}.\) Finally, the super edge-magic deficiency of \(G\), denoted by \(\mu_s(G)\), is the minimum non-negative integer \(n\) such that \(G \cup nK_1\), i.e., \(G\) together with \(n\) isolated vertices, is super edge-magic; defined to be \(\infty\), if such a \(\phi\) does not exist.NEWLINENEWLINEAs mentioned in this short note, the super edge-magic deficiency of a variety of graphs, including cycles, complete graphs, and complete bipartite graphs, have been identified. The authors of this note addressed this issue for some of the other graphs. An \((n, t)\)-kite is a combination of a \(n\)-cycle, \(C_n\), and a \(t\)-edge path, \(P_t\), when one vertex of \(C_n\) is adjacent to an end vertex of \(P_t.\) It is known that \((n, 2)\text{-kite}\) is super edge-magic if and only if \(n\) is even. The authors showed that, when \(n\) is odd, \(\mu_s((n, 2)\text{-kite})=1;\) and, by example, when \(t \equiv 0, 1 \pmod{4}\), \(\mu_s((n, t)\text{-kite}) \leq 1\); but, \(\mu_s((n, t)\text{-kite}) \leq 2\), when \(t \equiv 2, 3 \pmod{4}\).NEWLINENEWLINEThe graph \(C_n \odot P_t\) generalizes the \((n, t)\)-kite in the sense that each vertex of \(C_n\) is adjacent to an end vertex of a \(P_t\) path. It is also known that both \(C_n \odot P_2\) and \(C_n \odot P_3\) are super edge-magic for all odd \(n \geq 3;\) and the authors of this note showed that, by an example again, \(\mu_s(C_n \odot P_1) \leq 1\), if \(n \equiv 1\pmod{4}\); and \(\mu_s(C_n \odot P_1) \leq 2\), if \(n \equiv 2\pmod{4}\).NEWLINENEWLINEIt does look like a challenging problem to seek this interesting quantity for a general family of graphs.