Abstract

For a simple graph G consider an injection μ : V U E -> N. If for every vertex x in V we have μ(x) + Sum μ(xy) = h where y is adjacent to x, and for every edge xy in E we have μ(x) + μ(xy) + μ(y) = k, for some constants h and k, then μ is a totally magic injection (TMI) of G. Also, mt(G) is the smallest natural number such that there is a TMI μ : V U E --> {1, 2, . . . ,mt(G)}. Here we study TMIs and the number mt(G) for certain G. One theorem, the Star Theorem, is useful for eliminating many classes of well-known graphs that could have a TMI. For most n and nj the following graphs do not have a TMI: every non-star tree, Pn, Cn, Wn, Kn, and Kn1,n2,...,np . We determine mt(F) for every forest F that has a TMI, and mt(G) for every graph G with at most 6 vertices that has a TMI.

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