Graceful Signed Graphs
Mukti Acharya, Tarkeshwar Singh · Czechoslovak Mathematical Journal · 2004
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 + 2 d , ..., k + ( m − 1) d and − k , − ( k + d ), − ( k + 2 d ), ..., − ( 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.