On colorings of graph fractional powers

For any k∈Nk∈N, the kk-subdivision of graph GG is a simple graph G1k which is constructed by replacing each edge of GG with a path of length kk. In this paper we introduce the mth power of the nn-subdivision of GG, as a fractional power of GG, 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 GG is a connected graph with Δ(G)≥3Δ(G)≥3 and mn<1. It is also shown that this conjecture is true in some special cases.