Every 4-regular graph is acyclically edge-6-colorable

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamčik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that a(G) ≤ ∆ + 2 for any simple graph G with maximum degree ∆. Basavaraju and Chandran (2009) showed that every graph G with ∆ = 4, which is not 4-regular, satisfies the conjecture. In this paper, we settle the 4-regular case, i.e., we show that every 4-regular graph G has a(G) ≤ 6.