On a Conjecture Concerning the Friendly Index Sets of Trees

For a graph G = (V, E) and a binary labeling f : V(G) -> Z(2), let upsilon(f) (i) vertical bar f-1(i)vertical bar. The labling f is said to be friendly if vertical bar upsilon(f)(1) - upsilon(f)(0)vertical bar <= 1. Any vertex labeling f : V(G) -> Z(2) induces an edge labeling f * : E(G) -> Z(2) defined by f *(xy) = vertical bar f (x) - f (y)vertical bar. Let e(f)(i) = vertical bar f *(-1) (i)vertical bar. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) - {vertical bar e(f)(1) - ef(0)vertical bar : f is a friendly vertex labeling of G }. In [15] Lee and Ng conjectured that the friendly index sets of trees will form an arithmetic progression. This conjecture has been mentioned in [17] and other manuscripts. In this paper we will first determine the friendly index sets of certain caterpillars of diameter four. Then we will disprove the conjecture by presenting an infinite number of trees whose friendly index sets do not form an arithmetic progression.