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• W2964010127 abstract "A 1-factorization of a graph G is a collection of edge-disjoint perfect matchings whose union is E(G). A trivial necessary condition for G to admit a 1-factorization is that |V (G)| is even and G is regular; the converse is easily seen to be false. In this paper, we consider the problem of finding 1-factorizations of regular, pseudorandom graphs. Specifically, we prove that for any ϵ > 0, an (n, d, λ)-graph G (that is, a d-regular graph on n vertices whose second largest eigenvalue in absolute value is at most λ) admits a 1-factorization provided that n is even, C <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</sub> ≤ d ≤ n-1 (where C <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</sub> = C <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</sub> (∈) is a constant depending only on ∈), and λ ≤ d <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1-∈</sup> . In particular, since (as is well known) a typical random d-regular graph G <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n,d</sub> is such a graph, we obtain the existence of a 1-factorization in a typical G <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n,d</sub> for all C <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</sub> ≤ d ≤ n - 1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1-factorizations of such graphs G which is off by a factor of 2 in the base of the exponent from the known upper bound. This lower bound is better by a factor of 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>nd/2</sup> than the previously best known lower bounds, even in the simplest case where G is the complete graph. Our proofs are probabilistic and can be easily turned into polynomial time (randomized) algorithms." @default.
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• W2964010127 date "2018-10-01" @default.
• W2964010127 modified "2023-10-14" @default.
• W2964010127 title "1-Factorizations of Pseudorandom Graphs" @default.
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• W2964010127 doi "https://doi.org/10.1109/focs.2018.00072" @default.
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