PreprintCC BY 4.0arXiv
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.
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BibTeX
@misc{ma2026proof,
title = {{A proof of the 3-Decomposition Conjecture}},
author = {Jicheng Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.25385},
eprint = {2608.25385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25385},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A proof of the 3-Decomposition Conjecture
AU - Jicheng Ma
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.25385
UR - https://arxiv.org/abs/2608.25385
AB - 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.
ER -