Abstract
Gallucio, Goddyn and Hell proved in 2001 that in any minor-closed class, graphs with large enough girth have a homomorphism to any given odd cycle. In this paper, we study the computational aspects of this problem. We show that for any minor-closed class $F$ containing all planar graphs, and such that all minimal forbidden minors are 3-connected, the following holds: for any $k$ there is a $g = g(F, k)$ such that every graph of girth at least $g$ in $F$ has a homomorphism to $C_2k+1$, but deciding whether a graph of girth $g-1$ in $F$ has a homomorphism to $C_2k+1$ is NP-complete. The classes of graphs on which this result applies include planar graphs, $K_n$-minor free graphs, and graphs with bounded Colin de Verdière parameter (for instance, linklessly embeddable graphs). We also show that the same dichotomy occurs in problems related to a question of Havel (1969) and a conjecture of Steinberg (1976) about the 3-colorability of sparse planar graphs.