  For graphs $G$ and $H$, an $H$-coloring of $G$ is an edge-preserving mapping
from $V(G)$ to $V(H)$. Note that if $H$ is the triangle, then $H$-colorings are
equivalent to $3$-colorings. In this paper we are interested in the case that
$H$ is the five-vertex cycle $C_5$.
  A minimal obstruction to $C_5$-coloring is a graph that does not have a
$C_5$-coloring, but every proper induced subgraph thereof has a $C_5$-coloring.
In this paper we are interested in minimal obstructions to $C_5$-coloring in
$F$-free graphs, i.e., graphs that exclude some fixed graph $F$ as an induced
subgraph. Let $P_t$ denote the path on $t$ vertices, and let $S_{a,b,c}$ denote
the graph obtained from paths $P_{a+1},P_{b+1},P_{c+1}$ by identifying one of
their endvertices.
  We show that there is only a finite number of minimal obstructions to
$C_5$-coloring among $F$-free graphs, where $F \in \{ P_8, S_{2,2,1},
S_{3,1,1}\}$ and explicitly determine all such obstructions. This extends the
results of Kami\'nski and Pstrucha [Discr. Appl. Math. 261, 2019] who proved
that there is only a finite number of $P_7$-free minimal obstructions to
$C_5$-coloring, and of D\k{e}bski et al. [ISAAC 2022 Proc.] who showed that the
triangle is the unique $S_{2,1,1}$-free minimal obstruction to $C_5$-coloring.
  We complement our results with a construction of an infinite family of
minimal obstructions to $C_5$-coloring, which are simultaneously $P_{13}$-free
and $S_{2,2,2}$-free. We also discuss infinite families of $F$-free minimal
obstructions to $H$-coloring for other graphs $H$.
