Towards the Chen-Raspaud Conjecture

For an integer k , a homomorphism from a graph G to the Kneser graph K ( 2 k + 1 , k ) is equivalent to assigning to each vertex of G a k -subset of { 1 , … , 2 k + 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 2 k + 1 k and no odd cycles with fewer than 2 k + 1 vertices admits a homomorphism to K ( 2 k + 1 , k ). They also showed that the statement is true for k = 2. In this note we confirm the conjecture for k = 3.