On incidence coloring of graph fractional powers

For any $ n ∈ \mathbb{N} $, the n-subdivision of a graph $ G $ is a simple graph $ G^\frac{1}{n} $ which is constructed by replacing each edge of $ G $ with a path of length n. The m-th power of $ G $ is a graph, denoted by $ G^m $, with the same vertices of $ G $, where two vertices of $ G^m $ are adjacent if and only if their distance in $ G $ is at most m. In [M.N. Iradmusa, On colorings of graph fractional powers, Discrete Math. 310 (2010), no. 10-11, 1551-1556] the m-th power of the n-subdivision of $ G $, denoted by $ G^\frac{m}{n} $ is introduced as a fractional power of $ G $. The incidence chromatic number of $ G $, denoted by $ χ_i(G) $, is the minimum integer k such that $ G $ has an incidence k-coloring. In this paper, we investigate the incidence chromatic number of some fractional powers of graphs and prove the correctness of the incidence coloring conjecture for some powers of graphs.