Bipartite Turán number of paths and other trees

TL;DR

This work resolves the exact connected bipartite Turán numbers for long paths: for all integers $k\ge 3$ and bipartite part sizes with $b\ge a\ge k$, the maximum number of edges in a connected bipartite graph on $(a,b)$ avoiding a $P_{2k}$ (and also avoiding a $P_{2k-1}$) equals $(k-2)(b-1)+a$. The authors prove tight lower bounds via explicit constructions and establish matching upper bounds via induction that leverage vertex deletions preserving connectivity and balance. They extend the analysis to broom trees $S_{2k,d*1}$, obtaining exact values $ex_{b,c}(n,n,S_{2k,d})=ex_{b,c}(n,n,S_{2k+1,d})=nd$ for $n\ge d^2/2$ and $d>2k$, with additional upper bounds when the star is not too large relative to the path. The results advance understanding of bipartite Turán numbers in the connected setting and address open questions by Caro, Patkós and Tuza, with parallels from concurrent work by He, Salia and Zhu.

Abstract

We solve a recent question of Caro, Patkós and Tuza by determining the exact maximum number of edges in a bipartite connected graph as a function of the longest path it contains as a subgraph and of the number of vertices in each side of the bipartition. This was previously known only in the case where both sides of the bipartition have equal size and the longest path has size at most $5$. We also discuss possible generalizations replacing "path" with some specific types of trees.

Figures (1)

• Figure 1: The construction in Proposition \ref{['th:borneinf_general']}.

Theorems & Definitions (42)

• Definition 1.1: caro2025bipartite
• Proposition 1.2: Theorem 7 of caro2025bipartite
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3