The Extremal Function for Two Disjoint Cycles

Abstract

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order $$n\ge 9$$ and size at least $$\lfloor \frac{7n-13}{2}\rfloor $$ contains two disjoint theta graphs. We also show that every 2-edge-connected graph of order $$n\ge 6$$ and size at least $$3n-5$$ contains two disjoint cycles, such that any specified vertex with degree at least three belongs to one of them. The lower bound on size in both is sharp in general.