Graph triangle-free and maximun degree

Question

Let G=(V,E) be a graph trangle free.

a)Prove that $\Delta(G) \ge \frac{|V|}2$

b)Prove that there exists such a graph for |V|= 1,2

How would I prove this? particularly the part a)

thanks

Answer 1

a) is false.

Consider $C_n$, the cycle of length $n$ for $n\geq 5$. Note $C_n$ is $2$-regular (and triangle free), so $\Delta(C_n)=2$. Hence, there can be no vertex with degree $$\geq \frac{n}{2}\geq\frac {5}{2}=2.5.$$