Towards the Chen-Raspaud conjecture

Abstract

For an integer k, a homomorphism from a graph G to the Kneser graph K(2k+1,k) is equivalent to assigning to each vertex of G a k-subset of {1,…,2k+1} in a way that adjacent vertices receive disjoint subsets.Chen and Raspaud (2010) [5] conjectured that for every k≥2, every graph G with maximum average degree less than 2k+1k and no odd cycles with fewer than 2k+1 vertices admits a homomorphism to K(2k+1,k). They also showed that the statement is true for k=2. In this note we confirm the conjecture for k=3.