A localized approach for Turán number of long cycles

Abstract Let be a simple graph and denote the length of the longest cycle containing an edge if there exists a cycle containing , and 2 otherwise. We prove that the summation of 2 divided by over all edges in an ‐vertex graph is at most , and characterize all ‐vertex extremal graphs that achieve the bound, thereby extending the classic Erdős–Gallai theorem on cycles.