Graceful Signed Graphs

Construction of Graceful Signed Graphs

“…Then necessary condition for S to be graceful is that it is possible to partition V (G):= V (S) into two subsets V o and V e such that the numbers m While Theorem 1 gives a necessary condition for a sigraph S to be graceful, Theorem 2 claims the impossibility of having a characterisation of graceful sigraphs by means of forbidding a class of sigraphs to be the induced subsigraphs. To date, several families of graceful as well as non-graceful graphs (sigraphs) have been discovered 1,2,7,8,[10][11][12][13][14][15] towards gaining better insight into their general properties.…”

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

“…It was conjectured in [9] that the converse of Corollary 1.1 must also hold for all k 3. Further, the following result was obtained.…”

“…Theorem 2A [9]. If a heterogeneous signed cycle Z k of length k ≡ 0 (mod 4) is graceful then the number of negative sections of odd lengths in Z k is even.…”

“…The notion of graceful graphs in graph theory (see [3], [4], [6], [7], [11], [14], [15], [19], [22]- [24]) was recently extended to the class of sigraphs (see [8], [9]) as follows:…”

“…. , −n}; such a labelling f is called a graceful labelling of S. A sigraph which admits such a labelling is called a graceful sigraph (see [9]). If E − (S) = ∅ in the above definition one obtains the standard notion of graceful graphs and graceful numberings of a graph (see [14], [15], [22]).…”