Abstract
We consider the Johnson graph J( n, w), 0 ⩽ w ⩽ n. The graph has ( n w ) vertices representing the ( n w ) w-subsets of an n-set. Two vertices are connected by an edge if the intersection between their w-subsets is a ( w − 1)-set. Let θ( n, w) be the chromatic number of this graph. It is well known that θ( n, w) ⩽ n. We give some constructions which yield θ( n, w) < n for some cases of n and w. The colorings associated with the chromatic number and other colorings of the graph lead to improvements on the lower bounds on the sizes of some constant weight codes.
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