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 minimum number of simple cycles covering a vertex or an edge of this graph. Namely, we prove that X(G) ≤ 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 $\left[\frac{ek!}{2}-\frac{k+1}{2}\right]$ simple cycles. Tight bounds on the number of simple cycles covering an edge or a vertex of a k-critical graph are also proved.