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On the size of graphs labeled with a condition at distance two

✍ Scribed by Georges, John P.; Mauro, David W.

Publisher: John Wiley and Sons
Year: 1996
Tongue: English
Weight: 595 KB
Volume: 22
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199605)22:1<47::aid-jgt7>3.0.co;2-l

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✦ Synopsis

A labeling of graph G with a condition at distance two is an integer labeling of V(G) such that adjacent vertices have labels that differ by at least two, and vertices distance two apart have labels that differ by a t least one. The lambda-number of G, A(G), is the minimum span over all labelings of G with a condition a t distance two. Let G(n, k) denote the set of all graphs with order n and lambda-number k. In this paper, w e examine the sizes of graphs in G(n, k). We modify Chvatal's result on non-hamiltonian graphs to obtain a formula for the minimum size of a graph in G(n, k), and w e use an algorithmic approach to obtain a formula for the maximum size. Finally, w e show that for any integer j between the maximum and minimum sizes there exists a graph with size j in G(n, k).

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