On the number of edges in some graphs

Chunhui Lai

Published 2018-08-05Version 1

We prove that $\lim\inf_{n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{40}{99}},$ which is better than the previous bounds $\sqrt 2$ [Y. Shi, Discrete Math. 71(1988), 57-71], $\sqrt {2 + \frac{7654}{19071}}$ [C. Lai, Discrete Appl. Math. 232 (2017), 226-229]. We show that $\liminf_{n \rightarrow \infty} {g(n,m)-n\over \sqrt \frac{n}{m}} > \sqrt {2.444},$ for all even integer $m$.