On k-minimum and m-minimum edge-magic injections of graphs

An edge-magic total labelling (EMTL) of a graph G with n vertices and e edges is an injection λ : V (G)∪E(G)→ [n+e], where, for every edge uv ∈ E(G), we have wtλ(uv) = kλ, the magic sum of λ. An edgemagic injection (EMI) μ of G is an injection μ : V (G) ∪ E(G) → N with magic sum kμ and largest label mμ. For a graph G we define and study the two parameters κ(G): the smallest kμ amongst all EMI’s μ of G, and m(G): the smallest mμ amongst all EMI’s μ of G. We find κ(G) for G ∈ G for many classes of graphs G. We present algorithms which compute the parameters κ(G) and m(G). These algorithms use a G-sequence: a sequence of integers on the vertices of G whose