The Connected Bipartite Turán Problem for Long Cycles and Paths

TL;DR

This work determines the exact connected bipartite Turán numbers for forbidding long cycles and long paths with fixed color-class sizes, extending the classical Erd\H{o}s–Gallai/Kopylov framework to bipartite graphs. The authors develop a structural analysis of $2$-connected bipartite graphs with no cycle of length at least $2\ell$, combining Kopylov’s method with a refined Jackson-type lemma to fully characterize extremal configurations. They then deduce the connected bipartite Turán numbers for paths and long cycles, presenting precise extremal graphs $B_1(a,b,k)$ and $B_2(a,b,2\ell)$ and proving corollaries that rederive classical results (Gyárf{\a}s–Rousseau–Schelp; Jackson) in the bipartite, color-class–constrained setting. The results provide concise, unified proofs of these extremal numbers and establish a framework for further bipartite connected Turán problems, with potential applications to trees and related parameters; concurrent independent work is noted for overlapping questions. The approach highlights how prescribing color-class sizes yields rich, nontrivial extremal structures that balance connectivity and bipartite balance, often requiring degree-one or degree-two vertices in one part for optimality.

Abstract

Caro, Patkós, and Tuza initiated a systematic study of the bipartite Turán number for trees, and in particular asked for the extremal number of edges in connected bipartite graphs with prescribed color-class sizes that contain no paths of given lengths. In this paper, we determine these numbers exactly and describe all corresponding extremal configurations. Our approach first establishes a more general result for long cycles: we determine the exact structure of all 2-connected bipartite graphs with no cycle of length at least a given constant. The proof combines Kopylov's method for long cycles with a strengthened version of Jackson's classical lemma, in which every extremal configuration is characterized. To highlight the applicability of our results, we conclude with applications yielding concise proofs of classical theorems on bipartite Turán numbers, notably rederiving the results of Gyárfás, Rousseau, and Schelp for paths and Jackson for long cycles.

Figures (2)

• Figure 1: Extremal bipartite graphs $B_2(a,b,2\ell)$ (2-connected) and $B_1(a,b,k)$ (connected).
• Figure 2: Schematic structure of the extremal configuration arising in Jackson’s Lemma, showing the neighborhoods of the terminal vertices of a longest path in a 2-connected bipartite graph with circumference of length $2(d(v_1) + d(v_m) - 2)$.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Lemma 3: Jackson jackson1985long
• proof : Sketch of the proof
• proof : Proof of Theorem \ref{['Thm:2_connected_bipartite_Cycle_free']}
• proof : Proof of Theorem \ref{['Thm:Connected_Path_Bipartite']}
• Claim 1
• proof
• Theorem 4: Gyárfás, Rousseau, and Schelp gyarfas1984extremal
• proof : Proof using Theorem \ref{['Thm:Connected_Path_Bipartite']}