Further results on super edge-magic total labeling of extended \(w\)-trees (Q2831585)

Let \(G=(V(G),E(G))\) be a finite, simple, planar, undirected graph. A labeling of \(G\) is a map \(\lambda\) assigning vertices and edges of \(G\) to integers, and it is total, if its domain is given by \(V(G) \cup E(G)\). A total labeling of \(G\) is edge-magic, if there is an integer \(c\) such that \(\lambda(x)+\lambda(x,y)+\lambda(y)=c\) for all \(xy \in E(G)\), and it is super edge-magic if \(\lambda(V(G))=\{1,2, \ldots, |V(G)|\}\). This latter notion has been introduced by \textit{H. Enomoto} et al. [SUT J. Math. 34, No. 2, 105--109 (1998; Zbl 0918.05090)], who conjectured that every tree admits a super edge-magic total labeling. So far, this conjecture has been settled for particular classes of trees, in particular \(w\)-trees and extended \(w\)-trees. The authors prove some more results related to super edge-magic total labelings on extended \(w\)-trees, in particular disjoint unions of isomorphic and non-isomorphic copies of such trees.