The Connected Bipartite Tur\'an Problem for Long Cycles and Paths

TL;DR

This paper determines the exact extremal edge counts for connected bipartite graphs avoiding long cycles and paths, characterizing all extremal configurations and applying these results to classical bipartite Turán problems.

Contribution

It provides exact solutions and structural characterizations for bipartite Turán numbers related to long cycles and paths, extending previous partial results.

Findings

Exact extremal numbers for connected bipartite graphs with no long cycles or paths.

Structural descriptions of all extremal configurations.

Concise proofs of classical bipartite Turán theorems.

Abstract

Caro, Patk\'os, and Tuza initiated a systematic study of the bipartite Tur\'an 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 Tur\'an numbers, notably rederiving the results of Gy\'arf\'as, Rousseau, and Schelp for paths and Jackson for long cycles.