Above guarantee parameterization for vertex cover on graphs with maximum degree 4

In the vertex cover problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of vertices S of size at most k such that every edge of G is incident on at least one vertex in S. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by r = k -n/3. We give an algorithm for this problem whose running time is O*(1.6253r). As a corollary, we obtain an O*(1.2403k)-time algorithm for vertex cover on graphs with maximum degree 4.