1,254 research outputs found
Arithmetic progressions of cycles in outer-planar graphs
AbstractA question of Erdős asks if every graph with minimum degree 3 must contain a pair of cycles whose lengths differ by 1 or 2. Some recent work of Häggkvist and Scott (see Arithmetic progressions of cycles in graphs, preprint), whilst proving this, also shows that minimum degree 500k2 guarantees the existence of cycles whose lengths are m,m+2,m+4,…,m+2k for some m—an arithmetic progression of cycles. In like vein, we prove that an outer-planar graph of order n, with bounded internal face size, and outer face a cycle, must contain a sequence of cycles whose lengths form an arithmetic progression of length exp((clogn)1/3−loglogn). Using this we give an answer for outer-planar graphs to a question of Erdős concerning the number of different sets which can be achieved as cycle spectra
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
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
Popular progression differences in vector spaces II
Green used an arithmetic analogue of Szemer\'edi's celebrated regularity
lemma to prove the following strengthening of Roth's theorem in vector spaces.
For every , , and prime number , there is a least
positive integer such that if ,
then for every subset of of density at least there is
a nonzero for which the density of three-term arithmetic progressions with
common difference is at least . We determine for the
tower height of up to an absolute constant factor and an
additive term depending only on . In particular, if we want half the random
bound (so ), then the dimension required is a tower of
twos of height . It turns
out that the tower height in general takes on a different form in several
different regions of and , and different arguments are used
both in the upper and lower bounds to handle these cases.Comment: 34 pages including appendi
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