Characterizations of additively graceful signed paths and cycles

Abstract

A (p,m,n) signed graph S, is a signed graph of order p with m positive edges and n negative edges. In this paper, we first prove a few basic results on vertex labelings of paths. We use these results and a sequence of lemmas to obtain a characterization of additively graceful signed paths. We prove that, apart from exactly 4 exceptions, additively graceful signed paths are characterized by the signed paths containing at most one negative section with n less than or equal to 2. We also establish a characterization of additively graceful signed cycles. We prove that a (p,m,n) signed cycle S is additively graceful if and only if one among the following 4 conditions are satisfied, (a) n=0 and m identical to 0 or 3 (mod 4), (b) n = 1 and m identical to 1 or 2 (mod 4), (c) n = 2, or 2 (mod 4) and S contains a single negative section, (d) S is the all negative signed cycle on C sub(3).