Homomorphism bounds and edge-colourings of $K_4$-minor-free graphs

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.