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On Super Edge-Magic Graphs which are Weak Magic

Dharam Chopra1, Sin-Min Lee2
1Department of Mathematics and Statistics Wichita State University Wichita, KS 67260, USA
2Department of Computer Science San Jose State University San Jose, California 95192 U.S.A.

Abstract

A (p,q) graph G is total edge-magic if there exists a bijection f:VE{1,2,,p+q} such that for each e=(u,v)E, we have f(u)+f(e)+f(v) as a constant. For a graph G, denote M(G) the set of all total edge-magic labelings. The magic strength of G is the minimum of all constants among all labelings in M(G), denoted by emt(G). The maximum of all constants among M(G) is called the maximum magic strength of G and denoted by eMt(G).

Hegde and Shetty classify a magic graph as strong if emt(G)=eMt(G), ideal magic if 1eMt(G)emt(G)p, and \textbf{weak magic} if eMt(G)emt(G)>p. A total edge-magic graph is called a super edge-magic if f(V(G))={1,2,,p}. The problem of identifying which kinds of super edge-magic graphs are weak-magic graphs is addressed in this paper.