Lecture 4

1 Petersen’s Theorem Recall that a graph is cubic if every vertex has degree exactly 3, and bridgeless if it cannot be disconnected by deleting any one edge (i.e., 2-edge-connected). Theorem 1 (Petersen) Any bridgeless cubic graph has a perfect matching. Proof: We will show that for any V ⊆ U , we have o(G−U) ≤ |U | (here o(G) is the number of odd components of the graph G). The theorem will then follow from the Tutte-Berge formula. |δ(C1)| ≥ 3