Bipartite Turán number of trees
Yair Caro, Balázs Patkós, Zsolt Tuza
TL;DR
The paper advances bipartite Turán theory for trees by defining $ex_b(a,b,F)$, $ex_b(n,F)$ and their connected versions, and establishing general lower bounds and connections to classical Turán numbers via $ex(n,F)$ and $ex(2n,F)$. It delivers general bounds dependent on $\,Delta(F)$ and the bipartition of $F$, and provides exact or near-exact results for broad tree families, including all trees with at most six vertices, double stars, spiders, and the $(k,\ell)$-partition family of trees, also answering a question of Yuan and Zhang on $ex_b(n,\mathcal{T}_{k,\ell})$. The work introduces and leverages constructions, degree-based arguments, and the Bucić–Idzik–Tuza local switching technique to relate bipartite and non-bipartite Turán numbers. It further analyzes asymptotic worst-case ratios between Turán-type functions, establishing $\,gamma_b=2/3$ and deriving nontrivial lower bounds for $\,gamma_{b,c}$ and $\,gamma_{b\leftrightarrow c}$. These results broaden the understanding of extremal behavior in bipartite graphs and connect bipartite and classical Turán theories through precise constructions and bounds.
Abstract
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.