Prove or disprove: If graph $G$ has at most $3n-1$ edges, then $\chi(G) \leq 6$

Question

The question is to prove or disprove that if graph $G$ has at most $3n-1$ edges, then the chromatic number is at most $6$. I tried to work it out using the fact that $\chi(G) \leq \triangle(G) + 1$, but I can't think of a relation between the max degree and the number of edges.

Answer 1

Ok so, consider $K_7$, It has $21$ edges which $3*7$. But then add a path of length $2$ to it to get the graph $G$ like in the following picture:

Then $\chi(G)=7$, $n=9$and $E(G)=23 <3*n-1=26$. So dosent't work!