Coloring Reduced Kneser Graphs
نویسندگان
چکیده
The vertex set of a Kneser graph KG(m,n) consists of all n-subsets of the set [m] = {0, 1, . . . ,m − 1}. Two vertices are defined to be adjacent if they are disjoint as subsets. A subset of [m] is called 2stable if 2 ≤ |a − b| ≤ m − 2 for any distinct elements a and b in that subset. The reduced Kneser graph KG2(m,n) is the subgraph of KG(m,n) induced by vertices that are 2-stable subsets. We focus our study on the reduced Kneser graphs KG2(2n + 2, n). We achieve a complete analysis of its structure. From there, we derive that the circular chromatic number of KG2(2n + 2, n) is equal to its ordinary chromatic number, which is 4. A second application of the structural theorem shows that the chromatic index of KG2(2n + 2, n) is equal to its maximum degree except when n = 2.
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