Graceful Signed Graphs

Abstract

A (p, q)-sigraph S is an ordered pair (G, s) where G = (V, E) is a (p, q)-graph and s is a function which assigns to each edge of G a positive or a negative sign. Let the sets E ⁺ and E ⁻ consist of m positive and n negative edges of G, respectively, where m + n = q. Given positive integers k and d, S is said to be (k, d)-graceful if the vertices of G can be labeled with distinct integers from the set {0, 1, ..., k + (q − 1)d such that when each edge uv of G is assigned the product of its sign and the absolute difference of the integers assigned to u and v the edges in E ⁺ and E ⁻ are labeled k, k + d, k + 2d, ..., k + (m − 1)d and −k, − (k + d), − (k + 2d), ..., − (k + (n − 1)d), respectively.

In this paper, we report results of our preliminary investigation on the above new notion, which indeed generalises the well-known concept of (k, d)-graceful graphs due to B. D. Acharya and S. M. Hegde.