Disjoint Cycles in Planar and Triangle-free Graphs

Let \(p\) denote the number of vertices in a graph and let \(q\) denote the number of edges. Two cycles in a graph are disjoint if they have no common vertices. Pósa proved that any graph with \(q \geq 3p – 5\) contains two disjoint cycles. This result does not apply to planar graphs, since every planar graph has \(q \leq 3p – 6\).
In this paper, I show that any planar graph with \(q \geq 2p\) contains two disjoint cycles. I also show that this bound is best possible and that there is no minimum number of edges in a planar graph which will ensure the graph contains \(3\) disjoint cycles. Furthermore, a sufficient condition for any triangle-free graph (and therefore any bipartite graph) to contain \(k\) disjoint cycles is given.