Towards the Chen-Raspaud conjecture

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+1 k 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. (c) 2023 Elsevier B.V. All rights reserved.