The Extremal Function for Three Disjoint Theta Graphs

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 $n\ge 12$ and size at least$max\{\lceil \frac{3n+79}{2}\rceil ,\lfloor \frac{11n-33}{2}\rfloor \}$ contains three disjoint theta graphs. As a corollary, every graph of order $n\ge 12$ and size at least $max\{\lceil \frac{3n+79}{2}\rceil ,\lfloor \frac{11n-33}{2}\rfloor \}$ contains three disjoint cycles of even length. The lower bound on the size is sharp in general.