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