Sets of r-Graphs that Color All r-Graphs

An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping \(f:E(G) \rightarrow E(H)\) such that each r adjacent edges of G are mapped to r adjacent edges of H. For every \(r\ge 3\), let \(\mathcal H_r\) be an inclusion-wise minimal set of connected r-graphs, such that for every connected r-graph G there is an \(H \in \mathcal H_r\) which colors G. The Petersen Coloring Conjecture states that \(\mathcal H_3\) consists of the Petersen graph P. We show that if true, then this is a very exclusive situation. Our main result is that either \(\mathcal H_3 = \{P\}\) or \(\mathcal H_3\) is an infinite set and if \(r \ge 4\), then \(\mathcal H_r\) is an infinite set. In particular, for all \(r \ge 3\), \(\mathcal H_r\) is unique. We first characterize \(\mathcal H_r\) and then prove that if \(\mathcal H_r\) contains more than one element, then it is an infinite set. To obtain our main result we show that \(\mathcal H_r\) contains the smallest r-graphs of class 2 and the smallest poorly matchable r-graphs, and we determine the smallest r-graphs of class 2.