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 m th power of the n -subdivision of G , as a fractional power of G , 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 χ ( G m n ) = ω ( G m n ) provided that G is a connected graph with Δ ( G ) ≥ 3 and m n < 1 . It is also shown that this conjecture is true in some special cases.