  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 $n\ge 9,$ 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 $n\ge
10,$ there exists a cubic homogeneously traceable nonhamiltonian graph of order
$n;$ (2) for every integer $p\ge 18,$ there exists a $4$-regular homogeneously
traceable graph of order $p$ and circumference $p-4.$ Unsolved problems are
posed.
