A note on even disjoint union of paths

Abstract

A (p, q)-graph G is edge-magic if there exists a bijective function f: V(G) ≼ E(G) → {1, 2,…, p + q} such that f(u) + f(v) + f(uv) = k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f(V(G)) = {1, 2,…, p}.There is an interesting question to know the super edge-magicness of the even disjoint union of paths. In this paper we use an operation on digraphs that is in some sense a generalization of the Kronecker product of matrices and has a relation with (super) edge-magic graphs. In light of an operation on digraphs we solve partially the question.