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A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set {1,2,…,|V(G)|+|E(G)|}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{1,2,\ldots ,|V(G)|+|E(G)|\}$$\end{document} such that the sum f(x)+f(xy)+f(y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(x)+f(xy)+f(y)$$\end{document} for any xy in E(G) is constant. Such a function is called an edge-magic labeling of G, and the constant is called the valence of f. An edge-magic labeling with the extra property that f(V(G))={1,2,…,|V(G)|}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(V(G))= \{1,2,\ldots ,|V(G)|\}$$\end{document} is called super edge-magic. In this paper, we establish a relationship between the valences of (super) edge-magic labelings of certain types of bipartite graphs and the existence of a particular type of decompositions of such graphs.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Oct 22, 2020
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