On colorings of graph fractional powers

AbstractFor any k∈N, the k-subdivision of graph G is a simple graph G1k 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 Gmn. In this regard, we investigate the chromatic number and clique number of fractional power of graphs. Also, we conjecture that χ(Gmn)=ω(Gmn) provided that G is a connected graph with Δ(G)≥3 and mn