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.

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.

It is proved that if the points of the three-dimensional Euclidean space are coloured in red and blue, then there exist either two red points unit distance apart, or six collinear blue points with distance one between any two consecutive points.

An old question in Euclidean Ramsey theory asks, if the points in the plane are red-blue coloured, does there always exist a red pair of points at unit distance or five blue points in line separated by unit distances? An elementary proof answers this question in the affirmative.

No abstract