On acyclic edge-coloring of complete bipartite graphs

An acyclic edge-coloring of a graph is a proper edge-coloring without bichromatic (2colored) 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 ∆ = ∆(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 = ∆+ 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 = ∆+ 2 when p is an odd prime. The main tool in their approach is perfect 1-factorization ofKp,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 = ∆+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 possess a perfect 1-factorization using n + 2 = ∆ + 2 colors. In this general framework, we show that Kp2,p2 admits an acyclic edge-coloring using p + 2 colors and thus establishing a′(Kp2,p2) = p 2 + 2 = ∆ + 2 when p ≥ 5 is an odd prime.