On super edge-magicness and deficiencies of forests (Q2811819)

Functions on graphs is an interesting and insufficiently studied section of graph theory. As a rule, the function is defined on the set of vertices, and its value at each vertex must be reconciled with the values in the adjacent vertexes. These are a Grundy's function, a colorize function and some others. An edge-magic total labeling function is defined as a bijection mapping \(f\) from \(V(G)\cup E(G)\) to \(\{1,2,\dots,p+q\}\) (\(p\) is the quantity of vertexes, \(q\) is the quantity of edges), if there exists a constant \(k\) such that \(f(x)+f(xy)+f(y)=k\), for every edge \(xy\). The definition of this function is significantly different from the previously studied by the fact that this function is defined not only on the vertexes, but also on the edges of the graph. An edge-magic total labeling function \(f\) is called super if \(f(V(G))=\{1,2,\dots, p\}\). Continuing research starting by \textit{R. M. Figueroa-Centeno} et al. [Discrete Math. 231, No. 1--3, 153--168 (2001; Zbl 0977.05120)] the authors studied the super edge-magic total labeling and deficiency of forests consisting of combs, generalized combs and stars.NEWLINENEWLINEIt is clear that, in spite of the results obtained in this article, the general theory of such functions (at the level of criteria and algorithms) have not yet developed.