On colorings of graph fractional powers

For any k is an element of N, the k-subdivision of graph G is a simple graph G(1/k) which is constructed by replacing each edge of G with a path of length k. In this paper we introduce the mth power of the n-subdivision of G, as a fractional power of C. denoted by G(m/n). In this regard, we investigate the chromatic number and clique number of fractional power of graphs. Also, we conjecture that chi(G(m/n)) = omega(G(m/n)) provided that G is a connected graph with Delta(G) >= 3 and m/n < 1. It is also shown that this conjecture is true in some special cases. (c) 2010 Elsevier B.V. All rights reserved.