Counterexample to a conjecture of Györi on \(C_{2l}\)-free bipartite graphs (Q868335)

\textit{E. Györi} [Discrete Math. 165/166, 371--375 (1997; Zbl 0873.05058)] stated the following conjecture: Conjecture~1. If \(G=(X,Y)\) is a bipartite graph with color classes \(X\),\(Y\), where \(| X| =m\), \(| Y| =n\), \(m2\leq n\), \(3\leq l\leq m\) and \(G\) has at least \((l-1)n+m-l+2\) edges, then \(G\) must contain a cycle of length \(2l\). The current note provides a counterexample that disproves Conjecture~1 when \(m\geq 2l-1\). The following reformulation of Conjecture~1 is proposed: Conjecture~2. If \(G=(X,Y)\) is a bipartite graph with color classes \(X\),\(Y\), where \(| X| =m\), \(| Y| =n\), \(m2\leq n\), \(3\leq l\leq m\) such that \(m>(l-1)2\) and \(G\) has at least \((l-1)n+1/(l-1)m2\) edges, then \(G\) must contain a cycle of length \(2l\).