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LARGE $\mathcal {F}$-FREE SUBGRAPHS IN $r$-CHROMATIC GRAPHS

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  02 December 2022

ZHEN HE Open the ORCID record for ZHEN HE [Opens in a new window] ,
ZEQUN LV Open the ORCID record for ZEQUN LV [Opens in a new window]  and
XIUTAO ZHU Open the ORCID record for XIUTAO ZHU [Opens in a new window]
ZHEN HE
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China and MTA Rényi Institute, Budapest, Hungary e-mail: [email protected]
ZEQUN LV
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China and MTA Rényi Institute, Budapest, Hungary e-mail: [email protected]
XIUTAO ZHU*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China and MTA Rényi Institute, Budapest, Hungary
*
e-mail: [email protected]
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Abstract

For a graph G and a family of graphs $\mathcal {F}$, the Turán number ${\mathrm {ex}}(G,\mathcal {F})$ is the maximum number of edges an $\mathcal {F}$-free subgraph of G can have. We prove that ${\mathrm {ex}}(G,\mathcal {F})\ge {\mathrm {ex}}(K_r, \mathcal {F})$ if the chromatic number of G is r and $\mathcal {F}$ is a family of connected graphs. This result answers a question raised by Briggs and Cox [‘Inverting the Turán problem’, Discrete Math. 342(7) (2019), 1865–1884] about the inverse Turán number for all connected graphs.

Keywords

inverse Turán problemchromatic numberconnected graph

MSC classification

Primary: 05D05: Extremal set theory
Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 200 - 204
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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