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

#### Abstract

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.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.1.6

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