More odd graph theory from another point of view

S. Morteza Mirafzal

Published 2017-02-07Version 1

Let $ k$ be a positive integer with $ k \geq 2 $, and $ n= 2k+1$. We denote by $ O_k $ the graph with the $ k $ element subsets of $ I = \{ 1,2,...,n \} $ as vertices, where two such vertices are adjacent if they are disjoint. we roll call a graph $ \Gamma $ a $ VTNC$-graph, if it is a vertex transitive non-Cayley graph. In [7 ] it has been proved that $ O_k $ is a $ VTNC$-graph. The methods which used by the author to obtain the result, need some deep results from permutation group theory. In this paper we will show that $ O_k $ is a $ VTNC$-graph, by using rather elementary facts of number theory and group theory.