Reductions for the 3-Decomposition Conjecture

Abstract

The 3-decomposition conjecture is wide open. It asserts that every cubic graph can be decomposed into a spanning tree, a disjoint union of cycles, and a matching. We show that every such decomposition is derived from a homeomorphically irreducible spanning tree (HIST), which is surprising since a graph with a HIST trivially satisfies the conjecture.We also prove that the following graphs are reducible configurations with respect to the 3-decomposition conjecture: the Petersen graph with one vertex removed, the claw-square, the twin-house, and the domino. As an application, we show that all graphs of path-width at most 4 satisfy the 3-decomposition conjecture and that a 3-connected minimum counterexample to the conjecture is triangle-free, all cycles of length at most 6 are induced, and every edge is in the centre of an induced P6. Finally, employing a computer, we prove that all graphs of order at most 20 satisfy the 3-decomposition conjecture.

Automated testing and interactive construction of unavoidable sets for graph classes of small path‐width

“…is unavoidable for the class  3 of cubic graphs of path-width at most 3. To this end, let  P ( , ) be a smooth width-3 path-decomposition of a graph  ∈ G 3 . Let v and v′ be the vertices leaving the bags V 1 and V 2 , respectively.…”

“…These computational results led us to new insights on ξ k ( ). A further use of the algorithm can be found in [3,Lemma 29], where it is used to show that any cubic graph of path-width at most 4 contains one of the five specific graphs as a subgraph. This is used to prove the 3-decomposition conjecture for 3-connected cubic graphs of path-width at most 4.…”