GRACEFUL SIGNED GRAPHS


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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.

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