Cycle lengths in sparse graphs

TL;DR

This paper proves Erdos's conjecture on the number of cycle lengths in graphs with given girth and degree, and explores cycle lengths in graphs avoiding certain sequences, including powers of two.

Contribution

It confirms Erdos's conjecture on cycle length diversity in sparse graphs and provides bounds on cycle lengths avoiding specific sequences.

Findings

Proves that |C(G)| = Ω(d^{⌊(g-1)/2⌋}) for graphs with degree d and girth g.

Establishes bounds on the number of odd cycle lengths in graphs with given chromatic number and girth.

Provides upper bounds on the average degree of graphs avoiding cycles of certain lengths, including powers of two.

Abstract

Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =\Omega(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length \Omega(d^{\lfloor (g-1)/2\rfloor}). The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that \Omega(d^{\lfloor (g-1)/2\rfloor}) is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d. In the second part of the paper, motivated by the conjecture of Erdos and Gyarfas that every graph of minimum degree at least three contains a cycle of length a power of two, we prove a general theorem which gives an upper bound on the average degree of an n-vertex graph with no cycle of even length in a prescribed infinite sequence of integers. For many sequences, including the powers of two, our theorem gives the upper bound e^{O(\log^* n)} on the average degree of graph of order n with no cycle of length in the sequence, where \log^* n is the number of times the binary logarithm must be applied to n to get a number which is at most