Cycle length parities and the chromatic number

In 1966 Erdös and Hajnal proved that the chromatic number of graphs whose odd cycles have lengths at most l is at most l + 1. Similarly, in 1992 Gyárfás proved that the chromatic number of graphs which have at most k odd cycle lengths is at most 2k + 2 which was originally conjectured by Bollobás and Erdös. Here we consider the in influence of the parities of the cycle lengths modulo some odd prime on the chromatic number of graphs. As our main result we prove the following: Let p be an odd prime, k \in \mathbb{N} and I \subseteq {0,1,...,p-1} with \mid I \mid \leq p - 1. If G is a graph such that the set of cycle lengths of G contains at most k elements which are not in I modulo p, then X(G) \leq (1+ \frac{\mid I \mid}{p - \mid I \mid}) k+p (p-1)(r(2p,2p)+1)+1 where r(p,q) denotes the ordinary Ramsey number

Cycles with consecutive odd lengths

It is proved that there exists an absolute constant c > 0 such that for every natural number k, every non-bipartite 2-connected graph with average degree at least ck contains k cycles with consecutive odd lengths. This implies the existence of the absolute constant d > 0 that every non-bipartite 2-connected graph with minimum degree at least dk contains cycles of all lengths modulo k, thus providing an answer (in a strong form) to a question of Thomassen. Both results are sharp up to the constant factors.Comment: 7 page

Cycle lengths in sparse graphs

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 mos