For a graph G, let diam(G) denote the diameter of G. For any two vertices u and v in G, let d(u, v) denote the distance between u and v. A multilevel distance labeling (or distance labeling) for G is a function f that assigns to each vertex of G a non-negative integer such that for any vertices u and v, |f (u) − f (v)| ≥ diam(G) − d G (u, v) + 1.

We give a short proof for Chen's Alternative Kneser Coloring Lemma. This leads to a short proof for the Johnson-HolroydStahl conjecture that Kneser graphs have their circular chromatic numbers equal to their chromatic numbers.

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Abstract:The vertex set of the reduced Kneser graph KG 2 (m, 2) consists of all pairs {a, b} such that a, b 2 {1, 2, . . . , m} and 2 |a À b| m À 2. Two vertices are defined to be adjacent if they are disjoint. We prove that, if m ! 4 and m 6 ¼ 5, then the circular chromatic number of KG 2 (m, 2) is equal to m À 2, its ordinary chromatic number. ß