  Kronk introduced the $l$-path hamiltonianicity of graphs in 1969. A graph is
$l$-path Hamiltonian if every path of length not exceeding $l$ is contained in
a Hamiltonian cycle. We have shown that if $P=uvz$ is a 2-path of a
2-connected, $k$-regular graph on at most $2k$ vertices and $G - V(P)$ is
connected, then there must exist a Hamiltonian cycle in $G$ that contains the
2-path $P$. In this paper, we characterize a class of graphs that illustrate
the sharpness of the bound $2k$. Additionally, we show that by excluding the
class of graphs, both 2-connected, $k$-regular graphs on at most $2k + 1$
vertices and 3-connected, $k$-regular graphs on at most $3k-6$ vertices satisfy
that there is a Hamiltonian cycle containing the 2-path $P$ if $G\setminus
V(P)$ is connected.
