Subgroup regular sets in Cayley graphs |
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Institution: | 1. Rongcheng Campus, Harbin University of Science and Technology, Harbin, Heilongjiang 150080, People''s Republic of China;2. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia |
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Abstract: | Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an -regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in has exactly b neighbors in C. In particular, -regular sets and -regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an -regular set of G if there exists a Cayley graph of G which admits C as an -regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a -regular set of G, then it must also be an -regular set of G for any and such that a is even when is odd. A similar result involving -regular sets of such groups is also obtained in the paper. |
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Keywords: | Cayley graph Perfect code Regular set Perfect coloring Perfect 2-coloring |
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