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MAXIMUM CUTS IN GRAPHS WITHOUT WHEELS

Part of: Graph theory

Published online by Cambridge University Press:  26 December 2018

JING LIN Open the ORCID record for JING LIN [Opens in a new window] ,
QINGHOU ZENG Open the ORCID record for QINGHOU ZENG [Opens in a new window]  and
FUYUAN CHEN
JING LIN
Affiliation:
Center for Discrete Mathematics, Fuzhou University, Fujian 350003, China College of Mathematics and Physics, Fujian University of Technology, Fujian 350118, China email [email protected]
QINGHOU ZENG*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China email [email protected]
FUYUAN CHEN
Affiliation:
Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China email [email protected]
*
✉[email protected]
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Abstract

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For a graph $G$, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. Alon et al. [‘Maximum cuts and judicious partitions in graphs without short cycles’, J. Combin. Theory Ser. B 88 (2003), 329–346] conjectured that, for any fixed graph $H$, there exists an $\unicode[STIX]{x1D716}(H)>0$ such that $f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$. We show that, for any wheel graph $W_{2k}$ of $2k$ spokes, there exists $c(k)>0$ such that $f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$. In particular, we confirm the conjecture asymptotically for $W_{4}$ and give general lower bounds for $W_{2k+1}$.

Keywords

maximum cutH-freewheel graph

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C70: Factorization, matching, partitioning, covering and packing
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 1 , August 2019 , pp. 13 - 26
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The research of the first author is supported by Youth Foundation of Fujian Province (Grant No. JAT170398) and Foundation of Fujian University of Technology (Grant No. GY-Z15086); the research of the second author is supported by the Postdoctoral Science Foundation of China (Grant No. 2018M632528) and the Fundamental Research Funds for the Central Universities (Grant No. WK0010460005); the research of the third author is supported by NSFC (Grant No. 11601001).

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