On incidence coloring of graph fractional powers

For any n ∈ N, the n-subdivision of a graph G is a simple graph G 1n 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 Gm, with the same vertices of G, where two vertices of Gm 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 Gm n is introduced as a fractional power of G. The incidence chromatic number of G, denoted by χi(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.

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)