Some unsolved problems on cycles

TL;DR

This paper reviews unresolved problems in graph theory related to cycle decompositions, maximum edges in graphs avoiding certain subgraphs, and conjectures by Erdős, providing insights and skepticism about Hajos' conjecture.

Contribution

It summarizes existing results and conjectures on cycle decompositions, Turán numbers, and related unsolved problems, and expresses doubt about Hajos' conjecture.

Findings

Hajos' conjecture is likely false.

Provides bounds and partial results on cycle and Turán problems.

Discusses open problems and conjectures in graph theory.

Abstract

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 $ex(n,C_{2k})\geq cn^{1+1/k}$(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 Haj\'{o}s conjecture is true.