Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is Θ(n3.5logn)Θ(n3.5logn); but the best combinatorial   algorithms have approximation ratio 4/5 only, achieved in O(n2)O(n2) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477–504; E. Halperin, et al., J. Algorithms 53 (2004) 169–185]. Here we present an improved combinatorial approximation, which is a 5/6-approximation algorithm that runs in O(n2)O(n2) time, perhaps improvable even to O(n)O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3.