  Given a simple graph $G = (V, E)$ and a constant integer $k \ge 2$, the
$k$-path vertex cover problem ({\sc P$k$VC}) asks for a minimum subset $F
\subseteq V$ of vertices such that the induced subgraph $G[V - F]$ does not
contain any path of order $k$. When $k = 2$, this turns out to be the classic
vertex cover ({\sc VC}) problem, which admits a $\left(2 - {\rm
\Theta}\left(\frac 1{\log|V|}\right)\right)$-approximation. The general {\sc
P$k$VC} admits a trivial $k$-approximation; when $k = 3$ and $k = 4$, the best
known approximation results for {\sc P$3$VC} and {\sc P$4$VC} are a
$2$-approximation and a $3$-approximation, respectively. On $d$-regular graphs,
the approximation ratios can be reduced to $\min\left\{2 - \frac 5{d+3} +
\epsilon, 2 - \frac {(2 - o(1))\log\log d}{\log d}\right\}$ for {\sc VC} ({\it
i.e.}, {\sc P$2$VC}), $2 - \frac 1d + \frac {4d - 2}{3d |V|}$ for {\sc P$3$VC},
$\frac {\lfloor d/2\rfloor (2d - 2)}{(\lfloor d/2\rfloor + 1) (d - 2)}$ for
{\sc P$4$VC}, and $\frac {2d - k + 2}{d - k + 2}$ for {\sc P$k$VC} when $1 \le
k-2 < d \le 2(k-2)$. By utilizing an existing algorithm for graph defective
coloring, we first present a $\frac {\lfloor d/2\rfloor (2d - k + 2)}{(\lfloor
d/2\rfloor + 1) (d - k + 2)}$-approximation for {\sc P$k$VC} on $d$-regular
graphs when $1 \le k - 2 < d$. This beats all the best known approximation
results for {\sc P$k$VC} on $d$-regular graphs for $k \ge 3$, except for {\sc
P$4$VC} it ties with the best prior work and in particular they tie at $2$ on
cubic graphs and $4$-regular graphs. We then propose a $1.875$-approximation
and a $1.852$-approximation for {\sc P$4$VC} on cubic graphs and $4$-regular
graphs, respectively. We also present a better approximation algorithm for {\sc
P$4$VC} on $d$-regular bipartite graphs.
