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Graceful signed graphs

Mukti Acharya, Tarkeshwar Singh (2004)

Czechoslovak Mathematical Journal

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 u v of G is assigned the product of its sign and the absolute difference of the integers assigned to u and v the...

Graceful signed graphs: II. The case of signed cycles with connected negative sections

Mukti Acharya, Tarkeshwar Singh (2005)

Czechoslovak Mathematical Journal

In our earlier paper [9], generalizing the well known notion of graceful graphs, a ( p , m , n ) -signed graph S of order p , with m positive edges and n negative edges, is called graceful if there exists an injective function f that assigns to its p vertices integers 0 , 1 , , q = m + n such that when to each edge u v of S one assigns the absolute difference | f ( u ) - f ( v ) | the set of integers received by the positive edges of S is { 1 , 2 , , m } and the set of integers received by the negative edges of S is { 1 , 2 , , n } . Considering the conjecture therein that all...

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