On acyclic edge-coloring of complete bipartite graphs

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic (2-colored) cycles. The acyclic chromatic index of a graph G, denoted by a′(G), is the least integer k such that G admits an acyclic edge-coloring using k colors. Let Δ=Δ(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by Kn,n. Basavaraju, Chandran and Kummini proved that a′(Kn,n)≥n+2=Δ+2 when n is odd. Basavaraju and Chandran provided an acyclic edge-coloring of Kp,p using p+2 colors and thus establishing a′(Kp,p)=p+2=Δ+2 when p is an odd prime. The main tool in their approach is perfect 1-factorization of Kp,p. Recently, following their approach, Venkateswarlu and Sarkar have shown that K2p−1,2p−1 admits an acyclic edge-coloring using 2p+1 colors which implies that a′(K2p−1,2p−1)=2p+1=Δ+2, where p is an odd prime. In this paper, we generalize this approach and present a general framework to possibly get an acyclic edge-coloring of Kn,n which possesses a perfect 1-factorization using n+2=Δ+2 colors. In this general framework, using number theoretic techniques, we show that Kp2,p2 admits an acyclic edge-coloring with p2+2 colors and thus establishing a′(Kp2,p2)=p2+2=Δ+2 when p is an odd prime.