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- Title
Turán number for odd‐ballooning of trees.
- Authors
Zhu, Xiutao; Chen, Yaojun
- Abstract
The Turán number ex(n,H) $\text{ex}(n,H)$ is the maximum number of edges in an H $H$‐free graph on n $n$ vertices. Let T $T$ be any tree. The odd‐ballooning of T $T$, denoted by To ${T}_{o}$, is a graph obtained by replacing each edge of T $T$ with an odd cycle containing the edge, and all new vertices of the odd cycles are distinct. In this paper, we determine the exact value of ex(n,To) $\text{ex}(n,{T}_{o})$ for sufficiently large n $n$ and To ${T}_{o}$ being good, which generalizes all the known results on ex(n,To) $\text{ex}(n,{T}_{o})$ for T $T$ being a star, due to Erdős, Füredi, Gould, and Gunderson (1995), Hou, Qiu, and Liu (2018), and Yuan (2018), and provides some counterexamples with chromatic number three to a conjecture of Keevash and Sudakov (2004), on the maximum number of edges not in any monochromatic copy of H $H$ in a 2‐edge‐coloring of a complete graph of order n $n$.
- Subjects
RAMSEY numbers; COMPLETE graphs; TREES
- Publication
Journal of Graph Theory, 2023, Vol 104, Issue 2, p261
- ISSN
0364-9024
- Publication type
Article
- DOI
10.1002/jgt.22959