A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2- c o l o r e d ) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a ′ ( G ) . 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 K n , n . Alon, McDiarmid and Reed observed that a ′ ( K p − 1 , p − 1 ) = p for every prime p . In this paper we prove that a ′ ( K p , p ) ≤ p + 2 = Δ + 2 when p is prime. Basavaraju, Chandran and Kummini proved that a ′ ( K n , n ) ≥ n + 2 = Δ + 2 when n is odd, which combined with our result implies that a ′ ( K p , p ) = p + 2 = Δ + 2 when p is an odd prime. Moreover we show that if we remove any edge from K p , p , the resulting graph is acyclically Δ + 1 = p + 1 -edge-colorable.