The consecutively super edge-magic deficiency of graphs and related concepts

Rikio Ichishima (Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan University, 7-3-1 Nagayama, Tama-shi, Tokyo 206-8515, Japan)
Francesc A Muntaner-Batle (Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308 Australia)
Akito Oshima (Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308 Australia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 8, No 1 (2020): Electronic Journal of Graph Theory and Applications

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.