It is shown that for n ≥ 141, among all triangle-free graphs on n vertices, the balanced complete bipartite graph K ⌈n/2⌉,⌊n/2⌋ is the unique triangle-free graph with the maximum number of cycles. Using modified Bessel functions, tight estimates are given for the number of cycles in K ⌈n/2⌉,⌊n/2⌋ . Also, an upper bound for the number of Hamiltonian cycles in a triangle-free graph is given.

In this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that ∆ 4 = O(m), where ∆ is the maximal degree and m is the number of edges, the algorithm runs in expected time O(m). Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time O(m 2 ∆ 2 ) for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for powerlaw degree sequences in cases where the previous best was O(n 4.081 ), and (ii) O(nd + d 4 ) for d-regular graphs when d = o( √ n), where the previous best was O(nd 3 ).

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In this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that Δ4=O(m), where Δ is the maximal degree and m is the number of edges, the algorithm runs in expected time O(m). Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time O(m2Δ2) for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for power‐law degree sequences in cases where the previous best was O(n4.081), and (ii) O(nd + d4) for d‐regular graphs when d=o(n), where the previous best was O(nd3).

The main topic considered is maximizing the number of cycles in a graph with given number of edges. In 2009, Király conjectured that there is constant c such that any graph with m edges has at most (1.4) m cycles. In this paper, it is shown that for sufficiently large m, a graph with m edges has at most (1.443) m cycles. For sufficiently large m, examples of a graph with m edges and (1.37) m cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, exponentially tight bounds are proved for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.