A note on induced cycles in Kneser graphs

Abstract

Letg(n,r) be the maximal order of an induced cycle in the Knesser graph Kn([n] r), whose vertices are ther-sets of [n]={1, ...,n} and whose adjacency relation is disjointness. Thusg(n, r) is the largestm for which there is a sequenceA 1,A 2,...,A m υ [n] ofr-sets withA iA j=ϑ if and only if |i-j|=1 orm−1. We prove that there is an absolute constantc>0 for which

$$g(2r + 1,r) > c(2.587)^r $$

improving previous results. Our lower bound also shows that the clique covering number of the complement of ann-cycle is at most 1.459 log2 n for large enoughn. Related problems concerning the order of induced subgraphs of bounded degree of Kneser graphs are discussed.

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Research supported by FAPESP, Brazil, Proc. MAP 86/0904-8

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Kohayakawa, Y. A note on induced cycles in Kneser graphs. Combinatorica 11, 245–251 (1991). https://doi.org/10.1007/BF01205076

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 65