The $P_5$-free coloring conjecture for prime non-bipartite cores

Let $H$ be a graph. A prime graph is one admitting no non-trivial decomposition with respect to the direct product, a core is a graph with no homomorphism onto a proper subgraph, and a graph is $P_5$-free if it has no induced subgraph isomorphic to the path $P_5$. A minimal obstruction to $H$-coloring is a graph that does not admit a homomorphism to $H$, while every proper subgraph does. $P_5$-free coloring conjecture. If $H$ is a prime, non-bipartite core, then there is a finite family of minimal obstructions to $H$-coloring $P_5$-free graphs if and only if $H$ does not contain $C_4$ as a subgraph. The conjecture proposes a classification of when these minimal-obstruction families are finite; the paper notes that the corresponding behavior is already known for some odd cycles and cliques, but the general classification for prime targets remains open.