A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph K(n,k) are the subsets of {1,2,…,n} of cardinality k, two vertices are adjacent if and only if they are disjoint. The square G2 of a graph G is defined on the vertex set of G with two vertices adjacent if their distance in G is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when n=2k+1. It is believed that χ(K2(2k+1,k))=2k+c where c is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: 8k∕3+20∕3 for 1k≥3 (Kim and Park, 2014) and 32k∕15+32 for k≥7 (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to χ(K2(2k+1,k))≤5k∕2+c, where c is a constant in {5∕2,9∕2,5,6}, depending on k≥2.