  We present a necessary and sufficient condition for a graph of odd-girth
$2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least)
$2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph.
This yields a polynomial-time algorithm to recognize such bounds. Using this
condition, we first prove that every $K_4$-minor free graph of odd-girth $2k+1$
admits a homomorphism to the projective hypercube of dimension $2k$. This
supports a conjecture of the third author which generalizes the four-color
theorem and relates to several outstanding conjectures such as Seymour's
conjecture on edge-colorings of planar graphs. Strengthening this result, we
show that the Kneser graph $K(2k+1,k)$ satisfies the conditions, thus implying
that every $K_4$-minor free graph of odd-girth $2k+1$ has fractional chromatic
number exactly $2+\frac{1}{k}$. Knowing that a smallest bound of odd-girth
$2k+1$ must have at least ${k+2 \choose 2}$ vertices, we build nearly optimal
bounds of order $4k^2$. Furthermore, we conjecture that the suprema of the
fractional and circular chromatic numbers for $K_4$-minor-free graphs of
odd-girth $2k+1$ are achieved by a same bound of odd-girth $2k+1$. If true,
this improves, in the homomorphism order, earlier tight results on the circular
chromatic number of $K_4$-minor-free graphs. We support our conjecture by
proving it for the first few cases. Finally, as an application of our work, and
after noting that Seymour provided a formula for calculating the edge-chromatic
number of $K_4$-minor-free multigraphs, we show that stronger results can be
obtained in the case of $K_4$-minor-free regular multigraphs.
