On colorings of graph fractional powers

For any k ∈ 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 G, denoted by G m n . In this regard, we investigate chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(G m n ) = ω(G m n ) provided that G is a connected graph with ∆(G) ≥ 3 and m n < 1. It is also shown that this conjecture is true in some special cases.