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Maximum bipartite subgraphs in H -free graphs

Jing Lin

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 3, page 621-635
  • ISSN: 0011-4642

Abstract

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Given 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 . In this paper we prove that f ( m , θ k , s ) ≥ 1 2 m + Ω ( m ( 2 k + 1 ) / ( 2 k + 2 ) ) , which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write K k ' and K t , s ' for the subdivisions of K k and K t , s . We show that f ( m , K k ' ) ≥ 1 2 m + Ω ( m ( 5 k - 8 ) / ( 6 k - 10 ) ) and f ( m , K t , s ' ) ≥ 1 2 m + Ω ( m ( 5 t - 1 ) / ( 6 t - 2 ) ) , improving a result of Q. Zeng, J. Hou. We also give lower bounds on wheel-free graphs. All of these contribute to a conjecture of N. Alon, B. Bollobás, M. Krivelevich, B. Sudakov (2003).

How to cite

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Lin, Jing. "Maximum bipartite subgraphs in $H$-free graphs." Czechoslovak Mathematical Journal 72.3 (2022): 621-635. <http://eudml.org/doc/298359>.

@article{Lin2022,
abstract = {Given 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 $H$. In this paper we prove that $f(m,\theta _\{k,s\})\ge \tfrac\{1\}\{2\} m +\Omega (m^\{(2k+1)/(2k+2)\})$, which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write $K^\{\prime \}_\{k\}$ and $K^\{\prime \}_\{t,s\}$ for the subdivisions of $K_k$ and $K_\{t,s\}$. We show that $f(m,K^\{\prime \}_\{k\})\ge \tfrac\{1\}\{2\} m +\Omega (m^\{(5k-8)/(6k-10)\})$ and $f(m,K^\{\prime \}_\{t,s\})\ge \tfrac\{1\}\{2\} m +\Omega (m^\{(5t-1)/(6t-2)\})$, improving a result of Q. Zeng, J. Hou. We also give lower bounds on wheel-free graphs. All of these contribute to a conjecture of N. Alon, B. Bollobás, M. Krivelevich, B. Sudakov (2003).},
author = {Lin, Jing},
journal = {Czechoslovak Mathematical Journal},
keywords = {bipartite subgraph; $H$-free; partition},
language = {eng},
number = {3},
pages = {621-635},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Maximum bipartite subgraphs in $H$-free graphs},
url = {http://eudml.org/doc/298359},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Lin, Jing
TI - Maximum bipartite subgraphs in $H$-free graphs
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 3
SP - 621
EP - 635
AB - Given 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 $H$. In this paper we prove that $f(m,\theta _{k,s})\ge \tfrac{1}{2} m +\Omega (m^{(2k+1)/(2k+2)})$, which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write $K^{\prime }_{k}$ and $K^{\prime }_{t,s}$ for the subdivisions of $K_k$ and $K_{t,s}$. We show that $f(m,K^{\prime }_{k})\ge \tfrac{1}{2} m +\Omega (m^{(5k-8)/(6k-10)})$ and $f(m,K^{\prime }_{t,s})\ge \tfrac{1}{2} m +\Omega (m^{(5t-1)/(6t-2)})$, improving a result of Q. Zeng, J. Hou. We also give lower bounds on wheel-free graphs. All of these contribute to a conjecture of N. Alon, B. Bollobás, M. Krivelevich, B. Sudakov (2003).
LA - eng
KW - bipartite subgraph; $H$-free; partition
UR - http://eudml.org/doc/298359
ER -

References

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