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MAXIMUM CUTS IN GRAPHS WITHOUT WHEELS

Published online by Cambridge University Press:  26 December 2018

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 sd_frf@163.com
QINGHOU ZENG*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China email zengqh@ustc.edu.cn
FUYUAN CHEN
Affiliation:
Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui 233030, China email chenfuyuan19871010@163.com

Abstract

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 $W_{2k+1}$.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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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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