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Induced embeddings in Steinhaus graphs

✍ Scribed by Delahan, Franz A.

Publisher: John Wiley and Sons
Year: 1998
Tongue: English
Weight: 256 KB
Volume: 29
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199809)29:1<1::aid-jgt1>3.0.co;2-m

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✦ Synopsis

Fix any positive integer n. Let S be the set of all Steinhaus graphs of order n(n -1)/2 + 1. The vertices for each graph in S are the first n(n -1)/2 + 1 positive integers. Let I be the set of all labeled graphs of order n with vertices of the form i(i -1)/2 + 1 for the first n positive integers i. This article shows that the function φ : S → I that maps a Steinhaus graph to its induced subgraph is a bijection. Therefore, any graph of order n is isomorphic to an induced subgraph of a Steinhaus graph of order n(n -1)/2 + 1. This considerably tightens a result of Brigham, Carrington, and Dutton in [Brigham, Carrington, & Dutton, Combin. Inform. System Sci. 17 (1992)], which showed that this could be done with a Steinhaus graph of order 2 n-1 .

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