Independence Number and Disjoint Theta Graphs

The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a $\theta$-graph to be a pair of vertices $u, v$ with three internally disjoint paths joining $u$ to $v$. Given an independence number $\alpha$ and a fixed integer $k$, the results contained in this paper provide sharp bounds on the order $f(k, \alpha)$ of a graph with independence number $\alpha(G) \leq \alpha$ which contains no $k$ disjoint $\theta$-graphs. Since every $\theta$-graph contains an even cycle, these results provide $k$ disjoint even cycles in graphs of order at least $f(k, \alpha) + 1$. We also discuss the relationship between this problem and a generalized ramsey problem involving sets of graphs.