Abstract.
The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n−2k+2. Zoltán Füredi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that χ(K 2(2k+1, k))≤4k when k is odd and χ(K 2(2k+1, k))≤4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on χ(K 2(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.
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Acknowledgments.
The authors thank Z. Füredi for introducing this problem and providing helpful discussion and suggestions. The authors also thank A.V. Kostochka and D.B. West for their helpful suggestions and comments about this paper.
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This work was partially supported by NSF grant DMS-0099608
Final version received: April 23, 2003
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Kim, SJ., Nakprasit, K. On the Chromatic Number of the Square of the Kneser Graph K(2k+1, k). Graphs and Combinatorics 20, 79–90 (2004). https://doi.org/10.1007/s00373-003-0536-x
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DOI: https://doi.org/10.1007/s00373-003-0536-x