CN-TST 2014 Problem 11

Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$ , if $|E|\geq c|V|$ , then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.

Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$ , $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$ ; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$ , such that there exist $i,j, 1< i-j< n-1$ , satisfying $v_iv_j\in E$ .