# Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube

@article{Balogh2014UpperBO,
title={Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube},
author={J{\'o}zsef Balogh and Ping Hu and Bernard Lidick{\'y} and Hong Liu},
journal={Eur. J. Comb.},
year={2014},
volume={35},
pages={75-85}
}
• Published 31 December 2011
• Mathematics
• Eur. J. Comb.

## Figures and Tables from this paper

Tur\'an densities of hypercubes
• R. Baber
• Mathematics, Computer Science
• 2012
A number of extensions to Razborov’s semidefinite flag algebra method are described, including one which can be applied in a more general setting, notably to 3-uniform hypergraphs, to get a new upper bound of 0.5615 for π(K 4 ).
Minimal Saturated Subgraphs of the Hypercube
Within the hypercube Qn, we investigate bounds on the saturation number of a forbidden graph G, defined as the minimum number of edges in a subgraph H of Qn that is both G-free and has the property
Turán densities of hypercubes
• R. Baber
• Mathematics, Computer Science
• 2012
A number of extensions to Razborov's semidefinite flag algebra method are described, including one which can be applied in a more general setting, notably to 3-uniform hypergraphs, to get a new upper bound of 0.5615 for $\pi(K_4^3)$.
Generalized Tur\'an densities in the hypercube
• Mathematics
• 2022
A classical extremal, or Turán-type problem asks to determine ex(G,H), the largest number of edges in a subgraph of a graph G which does not contain a subgraph isomorphic to H . Alon and Shikhelman
On a Covering Problem in the Hypercube
• Mathematics
Graphs Comb.
• 2015
This paper addresses a particular variation of the Turán problem for the hypercube and gives upper and lower bounds for each of the questions and provides constructions of the set S above for some specific cases.
The $Q_2$-Free Process in the Hypercube
• Mathematics, Computer Science
Electron. J. Comb.
• 2020
The main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges.
Extremal problems on the hypercube
The hypercube, Qd, is a natural and much studied combinatorial object, and we discuss various extremal problems related to it. A subgraph of the hypercube is said to be (Qd, F )-saturated if it
A New Bound for the 2/3 Conjecture†
• Mathematics
Combinatorics, Probability and Computing
• 2013
We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours

## References

SHOWING 1-10 OF 32 REFERENCES
Subgraphs of a hypercube containing no small even cycles
Upper and lower bounds for the maximum number of edges in a subgraph of a hypercube containing no four-cycles or more generally, no 2k-cycles C2 are obtained.
Turán densities of hypercubes
• R. Baber
• Mathematics, Computer Science
• 2012
A number of extensions to Razborov's semidefinite flag algebra method are described, including one which can be applied in a more general setting, notably to 3-uniform hypergraphs, to get a new upper bound of 0.5615 for $\pi(K_4^3)$.
A note on short cycles in a hypercube
• Mathematics
Discret. Math.
• 2006
On the maximum number of edges in a c4-free subgraph of qn
• Mathematics
J. Graph Theory
• 1995
For the maximum number f(n) of edges in a C4-free subgraph of the n-dimensional cube-graph Qn is proved, which disproves one version of a conjecture of P. Erdos.
Highly Symmetric Subgraphs of Hypercubes
• Mathematics
• 1993
Two questions are considered, namely (i) How many colors are needed for a coloring of the n-cube without monochromatic quadrangles or hexagons? We show that four colors suffice and thereby settle a
A Ramsey‐type result for the hypercube
• Mathematics
J. Graph Theory
• 2006
We prove that for every fixed k and ℓ ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2 ℓ. This answers an open
On the number of pentagons in triangle-free graphs
• Mathematics
J. Comb. Theory, Ser. A
• 2013
On even-cycle-free subgraphs of the hypercube
• Mathematics
Electron. Notes Discret. Math.
• 2009
A New Lower Bound Based on Gromov’s Method of Selecting Heavily Covered Points
• Mathematics
Discret. Comput. Geom.
• 2012
Using methods from extremal combinatorics, one of the quantities appearing in Gromov’s approach is improved and thereby provided a new stronger lower bound on cd for arbitrary d, which is improved from 0.06332 to more than 0.07480.