Graceful signed graphs: II. The case of signed cycles with connected negative sections |
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Authors: | Mukti Acharya Tarkeshwar Singh |
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Affiliation: | (1) Dept of Applied Mathematics, Delhi College of Engineering, Bawana Road, 110042 Delhi, India |
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Abstract: | 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 uv 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 signed cycles Zk, of admissible length k 3 and signed structures, are graceful, we establish in this paper its truth for all possible signed cycles of lengths 0, 2 or 3 (mod 4) in which the set of negative edges forms a connected subsigraph. |
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Keywords: | graceful signed graphs signed cycles |
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