We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-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 K4-minor free graph of odd-girth 2k + 1 has fractional chromatic number exactly 2 + 1 k . Knowing that a smallest bound of odd-girth 2k + 1 must have at least k+2 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 K4-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 K4-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 K4-minor-free multigraphs, we show that stronger results can be obtained in the case of K4-minor-free regular multigraphs. *

A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs. 1

One of the De Bruijn -Erdős theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, nearpencils and finite projective planes. Chen and Chvátal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u, v, w} is a hyperedge. With this definition, the De Bruijn -Erdős theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result generalizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case. 1

a b s t r a c tA binary Cayley graph is a Cayley graph based on a binary group. In 1992, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd cycle, implying that such a graph cannot have chromatic number 3.We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homomorphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove a special case of this conjecture.