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.

On the super edge-magic deficiency of some graphs

“…Firstly, we show that for = 1, 2, 3, ( ,3 ) = 0. For = 1, 2, 3, label ( , ), ( 1 , ⋯ , ), ( 1 , 2 , 3 ) with (3, 2), (1), (5,4,6); (2,3), (1,4), (6,5,7); and (1, 4), (2,3,7), (5,8,6), respectively. These vertex labelings can be extended to a SEM labeling of ,3 for = 1, 2, 3.…”

“…For = 1, 2 and = 4, label ( , ), ( 1 , ⋯ , ), ( 1 , 2 , 3 , 4 ) with (3, 2), (1), (4,6,5,7) and (2,3), (1,4), (5,7,6,8), respectively. Moreover, for = 1, 2 and = 5, label ( , ), ( 1 , ⋯ , ), ( 1 , 2 , 3 , 4 , 5 ) with (2, 1), (3), (4,7,5,8,6) and (3,2), (1,4), (5,8,6,9,7), respectively. It can be checked that these labelings extend to a SEM labeling of ,4 and ,5 for = 1, 2.…”

“…The super edge-magic deficiency (SEMD) of a graph G , , is defined as either the minimum nonnegative n such that is a SEM graph or +∞ if there exists no such n . In 2016, Ichishima et al [7] give the notion of consecutively super edge-magic deficiency of a bipartite graph. The consecutively super edge-magic deficiency (consecutively SEMD) of a bipartite graph G , , is defined to be either the smallest nonnegative integer n with the property that is consecutively SEM or +∞ if there exists no such n .…”

On the (Consecutively) Super Edge-Magic Deficiency of Subdivision of Double Stars

“…is concept motivated Figueroa-Centeno et al [14] to introduce the concept of super edge-magic deficiency of a graph. e super edge-magic deficiency (SEMD) of a graph G, μ s (G), is defined as either the minimum nonnegative n such that G ∪ nK 1 is a SEM graph or +∞ if there exists no such n. Moreover, Ichishima et al [15] defined a similar notion for consecutively SEM labeling. e consecutively SEMD of a graph G, μ c (G), is defined to be either the smallest nonnegative integer n with the property that G ∪ nK 1 is consecutively SEM or +∞ if there exists no such n.…”

“…However, Conjecture 1 is still open. Meanwhile, in [15], Ichishima et al presented some results on consecutively SEMD of forets with two components, where its components are (non) isomorphic stars and union of paths and stars. In this paper, we study the (consecutively) SEMD of subdivision of double stars.…”