Regular homogeneously traceable nonhamiltonian graphs

A graph is called homogeneously traceable if every vertex is an endpoint of a Hamilton path. In 1979 Chartrand, Gould and Kapoor proved that for every integer n≥9, there exists a homogeneously traceable nonhamiltonian graph of order n. The graphs they constructed are irregular. Thus it is natural to consider the existence problem of regular homogeneously traceable nonhamiltonian graphs. We prove two results: (1) For every even integer n≥10, there exists a cubic homogeneously traceable nonhamiltonian graph of order n; (2) for every integer p≥18, there exists a 4-regular homogeneously traceable graph of order p and circumference p−4. Unsolved problems are posed.