On regular 2-path Hamiltonian graphs

Kronk introduced the $l$-path Hamiltonicity 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 $G-V\left(P\right)$ 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 $G-V\left(P\right)$ is connected.