Bipartite Turán number of trees
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909490945720320 |
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| author | Caro, Yair Patkós, Balázs Tuza, Zsolt |
| author_facet | Caro, Yair Patkós, Balázs Tuza, Zsolt |
| contents | We start a systematic investigation concerning bipartite Turán number for trees. For a graph $F$ and integers $1 \leq a \leq b$ we define:
$(i)$\quad $ex_b(a, b, F)$ is the largest number of edges that an $F$-free bipartite graph can have with part sizes $a$ and $b$. We write $ex_b(n, F)$ for $ex_b(n, n, F)$.
$(ii)$\quad $ex_{b,c}(a, b, F)$ is the largest number of edges that an $F$-free connected, bipartite graph can have with part sizes $a$ and $b$. We write $ex_{b,c}(n, F)$ for $ex{b,c}(n, n, F)$.
Both definitions are similar for a family $\mathcal{F}$ of graphs.
We prove general lower bounds depending on the maximum degree of $F$, as well as on the cardinalities of the two vertex classes of $F$.
We derive upper and lower bounds for $ex_b(n,F)$ in terms of $ex(2n,F)$ and $ex(n, F)$, the corresponding classical (not bipartite) Turán numbers.
We solve both problems for various classes of graphs, including all trees up to six vertices for any $n$, for double stars $D_{s ,t}$ if $a \geq f(s,t )$, for some families of spiders, and more.
We use these results to supply an answer to a problem raised by L. T. Yuan and X. D. Zhang [{\it Graphs and Combinatorics}, 2017] concerning $ex_b( n, \mathcal{T}_{k,\ell} )$, where $\mathcal{T}_{k,\ell}$ is the family of all trees with vertex classes of respective cardinalities $k$ and $\ell$.
The asymptotic worst-case ratios between Turán-type functions are also inverstigated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09052 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bipartite Turán number of trees Caro, Yair Patkós, Balázs Tuza, Zsolt Combinatorics We start a systematic investigation concerning bipartite Turán number for trees. For a graph $F$ and integers $1 \leq a \leq b$ we define: $(i)$\quad $ex_b(a, b, F)$ is the largest number of edges that an $F$-free bipartite graph can have with part sizes $a$ and $b$. We write $ex_b(n, F)$ for $ex_b(n, n, F)$. $(ii)$\quad $ex_{b,c}(a, b, F)$ is the largest number of edges that an $F$-free connected, bipartite graph can have with part sizes $a$ and $b$. We write $ex_{b,c}(n, F)$ for $ex{b,c}(n, n, F)$. Both definitions are similar for a family $\mathcal{F}$ of graphs. We prove general lower bounds depending on the maximum degree of $F$, as well as on the cardinalities of the two vertex classes of $F$. We derive upper and lower bounds for $ex_b(n,F)$ in terms of $ex(2n,F)$ and $ex(n, F)$, the corresponding classical (not bipartite) Turán numbers. We solve both problems for various classes of graphs, including all trees up to six vertices for any $n$, for double stars $D_{s ,t}$ if $a \geq f(s,t )$, for some families of spiders, and more. We use these results to supply an answer to a problem raised by L. T. Yuan and X. D. Zhang [{\it Graphs and Combinatorics}, 2017] concerning $ex_b( n, \mathcal{T}_{k,\ell} )$, where $\mathcal{T}_{k,\ell}$ is the family of all trees with vertex classes of respective cardinalities $k$ and $\ell$. The asymptotic worst-case ratios between Turán-type functions are also inverstigated. |
| title | Bipartite Turán number of trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.09052 |