Coloring the square of the Kneser graph KG(2k+1,k) and the Schrijver graph SG(2k+2,k)

The Kneser graph  KG(n,k) is the graph whose vertex set consists of all kk-subsets of an nn-set, and two vertices are adjacent if and only if they are disjoint. The Schrijver graph  SG(n,k) is the subgraph of KG(n,k) induced by all vertices that are 2-stable subsets. The square  G2G2 of a graph GG is defined on the vertex set of GG such that distinct vertices within distance two in GG are joined by an edge. The span  λ(G)λ(G) of GG is the smallest integer mm such that an L(2,1)L(2,1)-labeling of GG can be constructed using labels belonging to the set {0,1,…,m}{0,1,…,m}. The following results are established. (1) χ(KG2(2k+1,k))⩽3k+2 for k⩾3k⩾3 and χ(KG2(9,4))⩽12; (2) χ(SG2(2k+2,k))=λ(SG(2k+2,k))=2k+2 for k⩾4k⩾4, χ(SG2(8,3))=8, λ(SG(8,3))=9, χ(SG2(6,2))=9, and λ(SG(6,2))=8.