Regular graphs of large girth and arbitrary degree

Abstract

For every integer d≥10, we construct infinite families {G n } n∈ℕ of d+1-regular graphs which have a large girth ≥log d |G n |, and for d large enough ≥1.33 · log d |G n |. These are Cayley graphs on PGL 2(F q ) for a special set of d+1 generators whose choice is related to the arithmetic of integral quaternions. These graphs are inspired by the Ramanujan graphs of Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is a prime. When d is not equal to the power of an odd prime, this improves the previous construction of Imrich in 1984 where he obtained infinite families {I n } n∈ℕ of d + 1-regular graphs, realized as Cayley graphs on SL 2(F q ), and which are displaying a girth ≥0.48·log d |I n |. And when d is equal to a power of 2, this improves a construction by Morgenstern in 1994 where certain families {M n } nN of 2k +1-regular graphs were shown to have girth ≥2/3·log2 k|M n |.

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Correspondence to Xavier Dahan.

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Supported by the GCOE Project “Math-for-Industry” of Kyushu University

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Dahan, X. Regular graphs of large girth and arbitrary degree. Combinatorica 34, 407–426 (2014). https://doi.org/10.1007/s00493-014-2897-6

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Mathematics Subject Classification (2000)

  • 05C25
  • 05C38