On the connection between the chromatic number of a graph and the number of cycles, covering a vertex or an edge

We prove several tight bounds on the chromatic number of a graph in terms of the minimal number of simple cycles, covering a vertex or an edge of this graph. Namely, we prove that $\chi(G)\leq k$ in the following two cases: any edge of $G$ is covered by less than $[e(k-1)!-1]$ simple cycles or any vertex of $G$ is covered by less than $[\frac{ek!}2-\frac{k+1}2]$ simple cycles. Tight bounds on the number of simple cycles covering an edge or a vertex of a $k$-critical graph are also proved.