The consecutively super edge-magic deficiency of graphs and related concepts - Rikio Ichishima; Francesc A Muntaner-Batle; Akito Oshima (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2020)
A bipartite graph G with partite sets X and Y is called consecutively super edge-magic if there exists a bijective function f : V(G) ⋃ E(G) → 1,2,...,|V(G)| + |E(G)| with the property that f(X) = 1,2,...,|X|, f(Y) = |X|+1, |X|+2,...,|V(G)| and f(u)+f(v) +f(uv) is constant for each uv ∈ E(G). The question studied in this paper is for which bipartite graphs it is possible to add a finite number of isolated vertices so that the resulting graph is consecutively super edge-magic. If it is possible for a bipartite graph G, then we say that the minimum such number of isolated vertices is the consecutively super edge-magic deficiency of G; otherwise, we define it to be +∞. This paper also includes a detailed discussion of other concepts that are closely related to the consecutively super edge-magic deficiency.