Some unsolved problems on cycles

الأصل · EN

Hajos' conjecture that every simple even graph on n vertices can be decomposed into at most (n-1)/2 cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 231 - 236). Let f(n) be the maximum number of edges in a graph on n vertices in which no two cycles have the same length. P. Erdos raised the problem of determining f(n) (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). Given a graph H, what is the maximum number of edges of a graph with n vertices not containing H as a subgraph? This number is denoted ex(n,H), and is known as the Turan number. P. Erdos conjectured that there exists a positive constant c such that ex(n,C₂ₖ)≥ cn¹⁺¹/ᵏ(see P. Erdos, Some unsolved problems in graph theory and combinatorial analysis, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 97--109, Academic Press, London, 1971). This paper summarizes some results on these problems and the conjectures that relate to these. We do not think Hajós conjecture is true.