Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T21:08:19.929Z Has data issue: false hasContentIssue false

A Note on Bipartite Graphs Without 2k-Cycles

Published online by Cambridge University Press:  11 October 2005

ASSAF NAOR
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA (e-mail: anaor@microsoft.com)
JACQUES VERSTRAËTE
Affiliation:
Faculty of Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Canada N2L 3G1 (e-mail: jverstra@math.uwaterloo.ca)

Abstract

The question of the maximum number $\mbox{ex}(m,n,C_{2k})$ of edges in an m by n bipartite graph without a cycle of length 2k is addressed in this note. For each $k \geq 2$, it is shown that $\mbox{ex}(m,n,C_{2k}) \leq \begin{cases} (2k-3)\bigl[(mn)^{\frac{k+1}{2k}} + m + n\bigr] & \mbox{ if }k \mbox{ is odd,}\\[2pt] (2k-3)\bigl[m^{\frac{k+2}{2k}}\, n^{\frac{1}{2}} + m + n\bigr] & \mbox{ if }k \mbox{ is even.}\\ \end{cases}$

Type
Paper
Copyright
2005 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)