Construction of 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 $n\ge 9,$ 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. In 2022 Hu and Zhan proved that for every even integer $n\ge 10$, there exists a cubic homogeneously traceable nonhamiltonian graph of order $n$, and for every integer $n\ge 18$, there exists a 4-regular homogeneously traceable nonhamiltonian graph of order $n$. They also posed the problem: Given an integer $k\ge 4$, determine the integers $n$ such that there exists a $k$-regular homogeneously traceable nonhamiltonian graph of order $n$. We prove two results: (1) For every odd integer $k\ge 5$, integer $p\ge 6$ and integer $q\in \{0,2,4,6\}$, there exists a $k$-regular homogeneously traceable nonhamiltonian graph of order $p(k-1)+q$; (2) for every even integer $k\ge 6$, integer $p\ge 6$ and integer $q\in \{0,1,2,3,4,5,6\}$, there exists a $k$-regular homogeneously traceable nonhamiltonian graph of order $p(k-1)+q$.