Abstract—For integers   k  ≥2 and  n ≥2k+1, let   ex  (  n  , C_2 k) den ote the maximum number of edges in a C_2k-free graph of order n, and EX (n, C_2k) denote the set of all graphs with ex (n, C_2k) edges. For k=3, it is well known that ex (n, C_2k) > 0.5338n^1+1/k for some large n. In this paper, we study the values of ex (n, C_2k) for k≥3 when n is small, their lower bounds were given based the three graphs without C_2k. The known result shows that it is the tight lower bound for k=3 and n=28, and we further conjecture that ex (n, C_2k)=(2  k   +1)(2 k −1)(2  k −2)/2 for  n= 4k2−  2k−2 and k≥4.

Index Terms—Extremal graph, even cycle, lower bounds, Turán numbers.

The authors are with Beijing Jiaotong University, China (e-mail: yqsun@bjtu.edu.cn).

Cite: Rui Zhang, Yongqi Sun, and Yali Wu, "On the Turán Numbers for Even Cycles," International Journal of Applied Physics and Mathematics vol. 4, no. 3, pp. 211-214, 2014.