On magic graphs
Abstract
A labelling of a graph G is an assignment f of labels to either the vertices or the edges of G that induces for each edge uυ in the former a label depending on the vertex labels f(u) and f(υ) and in the latter for each vertex u a label depending on the labels of the edges incident with it. By a (1,1) edge-magic labelling of a graph G(V,E) , we mean a bijection f from V ∪ E to {1,2,... ,|V ∪ E|} such that for all edges uυ ∈ E(G) , the value of f(u) + f(υ) + f(uυ) is the same. It is said to be (1,1) edge-antimagic if f(u) + f(υ) + f(uυ) are distinct for all uυ ∈ E(G) . We introduce several other variations of magic labellings and discuss what are called (1,1) vertex-magic, (1,1) vertex-antimagic, (1,0) edge-magic, (1,0) edge-antimagic, (1,0) vertex-magic, (1,0) vertex-antimagic, (0,1) edge-magic,(0,1) edge-antimagic, (0,1) vertex-magic and (0,1) vertex-antimagic graphs. We exhibit such magic and anti-magic labellings for a number of classes of graphs. We also derive several general results governing these graphs. We also raise some open problems and new conjectures.
