On Regular Factors in Regular Graphs with Small Radius

In this note we examine the connection between vertices of high eccentricity and the existence of k-factors in regular graphs. This leads to new results in the case that the radius of the graph is small (≤ 3), namely that a d-regular graph G has all k-factors, for k|V (G)| even and k ≤ d, if it has at most 2d+2 vertices of eccentricity > 3. In particular, each regular graph G of diameter ≤ 3 has every k-factor, for k|V (G)| even and k ≤ d.