Skolem graceful signed graphs
Abstract
A signed graph (in short, sigraph) is a graph each of whose edges is designated to be either positive or negative whence a graph may be regarded as a sigraph in which all the edges are positive. By a (p, m, n)-sigraph, we mean the sigraph S = (V(S), E(S)) with p vertices, m positive edges and n negative edges. A sigraph S is said to be Skolem graceful whenever there is a bijective function ψ : V(S) → {1, 2,...,p} such that the induced edge function gψ(uv) = s(uv)|ψ(u) - ψ(v)|, ∀uv ∈ E(S) assigns the numbers 1, 2,..., m to the positive edges and -1, -2,..., -n to the negative edges of S. In this paper, we report the results of our preliminary investigation on this new notion of Skolem gracefulness of a sigraph. In particular, we determine the sigraphs of the form mK2+ ∪ nK2- which are Skolem graceful.
