Antimagicness of star forests

Abstract

An edge labeling of a graph G is a bijection f : E(G) {1, 2, " " ", 1E(G)1}. The induced vertex sum f+ of f is a function defined on V(G) given by f+ (u) = E(G) f(uv) for all u E V(G). The value f+(u) is called the vertex sum at u. The graph G is called antimagic if there exists an edge labeling of G such that the vertex sums at all vertices of G are distinct. A star forest is the union of disjoint stars. Let S denote a star with n edges, and mS denote a star forest consisting of the disjoint m copies of S. It is known that mS2 is antimagic if and only if m = 1. In this study, a necessary condition and a sufficient condition are obtained whereby a star forest mS2 U Snr U Snz U " " " U Sn, (ni, n2, " " ", nk > 3) may be antimagic. In addition, a necessary and sufficient condition is obtained whereby a star forest mS2 US (n > 3) may be antimagic. Moreover, a graph consisting of disjoint stars together with an extra disjoint path is also verified to be antimagic.