Matching complements in subcubic graphs and a proof of the 3-Decomposition Conjecture

Abstract

We prove the 3-Decomposition Conjecture: every finite connected cubic loopless multigraph decomposes into a spanning tree, a 2-regular subgraph, and a matching. The proof rests on a new theorem on matching complements in subcubic graphs. Let H be a finite connected bridgeless simple graph of maximum degree three, and let S be its set of degree-two vertices, with |S| = k>= 2. We show that H has a matching of size equal to its cyclomatic number, |E(H)| - |V(H)| + 1, whose deletion leaves a single tree containing all of S, together with cycles disjoint from S. The proof is by induction on k, using an alternating-path exchange that stops at the first entry into the growing tree component.