Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Ma, Jicheng
Institution: Renmin University of China
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204479740