High Girth Cubic Graphs Map to the Clebsch Graph

We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1). Hopkins and Staton [8] and Bondy and Locke [2] proved that every (sub)cubic graph of girth at least 4 has an edge-cut containing at least 4 5 of the edges. The existence of such an edge-cut follows immediately from the existence of a 5-edge-coloring as described above, so our theorem may be viewed as a kind of coloring extension of their result (under a stronger girth assumption). Every graph which has a homomorphism to a cycle of length five has an above-described 5-edge-coloring; hence our theorem may also be viewed as a weak version of Nešetřil’s Pentagon Problem: Every cubic graph of sufficiently high girth maps to C5.