Academic Report of the School of Mathematical Sciences [2025] No. 096  

(High-Level University Construction Series Report No. 1118)  

Lecture Title: Turán Number of Books in Non-Bipartite Graphs  

Lecturer: Professor Liu Ruifang (Zhengzhou University)  

Lecture Time: October 15, 2025, 13:30–14:30  

Venue: Room 501, Huixing Building  

Abstract:  

Let \( ex(n, H) \) denote the Turán number of a given graph \( H \). A graph is color-critical if removing any single edge reduces its chromatic number. Simonovits’ chromatic critical edge theorem states: If \( H \) is color-critical with \( \chi(H) = k+1 \), then there exists \( n_0(H) \) such that \( ex(n, H) = e(T_{n,k}) \) (where \( T_{n,k} \) is the \( k \)-partite Turán graph), and \( T_{n,k} \) is the unique extremal graph for all \( n \geq n_0(H) \). A book graph \( B_{r+1} \) consists of \( r+1 \) triangles sharing a common edge (here, \( r \geq 0 \) is an integer). \( B_{r+1} \) is color-critical with \( \chi(B_{r+1}) = 3 \). By Simonovits’ theorem, the bipartite Turán graph \( T_{n,2} \) is the sole extremal graph for \( B_{r+1} \)-free graphs when \( n \) is sufficiently large. Edwards (independently Khadživanov and Nikiforov) later confirmed Erdős’ booksize conjecture, showing \( ex(n, B_{r+1}) = e(T_{n,2}) \) for \( n \geq n_0(B_{r+1}) = 6r \).  

Since \( T_{n,2} \) is bipartite, we study the Turán problem for **non-bipartite** \( B_{r+1} \)-free graphs of order \( n \). For \( r = 0 \) (triangle-free graphs), Erdős proved: If \( G \) is a non-bipartite triangle-free graph on \( n \) vertices, then \( e(G) \leq \left\lfloor \frac{(n-1)^2}{4} \right\rfloor + 1 \). For general \( r \geq 1 \), we determine the exact Turán number of \( B_{r+1} \) in non-bipartite graphs and characterize all extremal graphs for sufficiently large \( n \).  

Lecturer’s Biography:  

Professor Liu Ruifang is a Level 3 Professor and Doctoral Supervisor at the School of Mathematics and Statistics, Zhengzhou University, and a High-Level Talent in Henan Province. She holds titles including: Leading Talent in Basic Research, Zhongyuan Talent Program; Henan Provincial Distinguished Young Scholar; Henan Provincial Excellent Young Scholar; Academic and Technological Leader, Henan Provincial Department of Education; Young Core Teacher, Henan Higher Education Institutions; Supervisor of Excellent Master’s Theses, Henan Province. She serves as: Standing Committee Member, Graph Theory, Combinatorics and Applications Professional Committee, Chinese Society for Industrial and Applied Mathematics; Standing Director, Henan Operations Research Society; Standing Director, Henan Society for Optimization, Overall Planning and Methodology. Her research focuses on spectral graph theory and algebraic graph theory. She has published over 60 SCI papers in top journals such as European Journal of Combinatorics, Advances in Applied Mathematics, Electronic Journal of Combinatorics, Journal of Algebraic Combinatorics, and Discrete Mathematics. She has led 3 National Natural Science Foundation of China (NSFC) projects: 2 General Programs and 1 Youth Program. Additionally, she leads 1 Zhongyuan Talent Program Project, 1 Henan Provincial Distinguished Young Scholar Project, and 1 Henan Provincial Excellent Young Scholar Project. She has conducted academic visits at the Department of Mathematics, West Virginia University (USA), and the Department of Mathematics, Hong Kong Baptist University.  

Faculty and students are welcome to attend!  

Inviter: Huang Zejun  

School of Mathematical Sciences  

October 13, 2025