The Maximum Number of Triangles in Graphs Without Cycles of Length 0mod5

For a graph and a graph family , let denote the maximum number of copies of in an ‐free ‐vertex graph. Let . Bai, Tompkins, and Well conjectured that is attained if and each block of the graph is a . In this paper, we determine the exact value of and the extremal graphs for all . The novelty of our proof is to give a proper partition of the set of triangles in an extremal graph. On the basis of this partition, we obtain the partition of the edge set and thus the structure of an extremal graph. Our new method can also be applied to obtain some meaningful results in other settings.