A Bound on the Number of Edges in Graphs Without an Even Cycle

Abstract

We show that, for each fixed k, an n-vertex graph not containing a cycle of length 2k has at most 80

Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs

“…We introduce some definitions that are needed in the rest of our proof where we establish a lower bound on p(H) and combine it with the upper bound in (7).…”

“…In particular, for graphs (i.e., 2-uniform hypergraphs) the C 2 -free condition does not impose any restriction, and there is no difference between a (Berge) cycle C l and a linear cycle C lin l . Bondy and Simonovits [5] showed that for k ≥ 2, ex(n, C 2k ) ≤ c k n 1+ 1 k for all sufficiently large n. Improvements to the constant factor c k are made in [22,18,7]. The girth of a graph is the length of a shortest cycle contained in the graph.…”

Hypergraphs with Few Berge Paths of Fixed Length between Vertices

“…A general upper bound of ex(n, C 2k ) = O k (n 1+1/k ) was first published by Bondy and Simonovits [3]. Since then, improvements have been made to the multiplicative constant [20,19,7], and constructions have been found showing that the order of magnitude is correct for k ∈ {2, 3, 5} [4,11,21,1]. However, besides C 4 , C 6 , and C 10 , the order of magnitude is unknown.…”

Anti-Ramsey Numbers of Paths and Cycles in Hypergraphs

“…For paths, Simonovits and Sós [41] proved that ar(n, P 2t+3+ǫ ) = tn − t−1 2 + 1 + ǫ for large n, where ǫ = 0, 1 and P k is a path on k vertices. Comparing with the Turán number of paths ex(n, P k ) ≤ (k − 2)n/2 (2) given by Erdős and Gallai [10], it follows that ar(n, P k ) = ex(n, P k−1 ) + O(1) when k is odd, and ar(n, P k ) = ex(n, P k−2 ) + O(1) when k is even. For a cycle C k of order k, Erdős, Simonovits and Sós [11] conjectured that ar(n, C k ) = n k−2 2 + 1 k−1 + O(1).…”