# Is it $NP$-complete to decompose bridgeless cubic bipartite graph into edge-disjoint paths of length 3?

Motivated by this post on cubic graphs decompositions, I am interested in decomposing a connected bridgeless graph into edge-disjoint paths of length 3 (P4). My intuition is that it should be NP-complete but did not find a reduction.

I am aware that it is $NP$-complete to decide whether a cubic bipartite planar graph is decomposable into vertex-disjoint paths of length 2 (P3).

Is it $NP$-complete to decide whether a bridgeless cubic bipartite graph is decomposable into edge-disjoint paths of length 3 (P4)?

In a related note, Barnette conjecture states that every 3-connected cubic bipartite planar graph is Hamiltonian. This is equivalent to decomposing every such graph into Hamiltonian cycle and a perfect matching. Feder and Subi proved that if there is a single graph in the class of the conjecture which does not admit such decomposition then deciding the existence of Hamiltonian cycle in $NP$-complete in that class.

For general connected cubic graph decomposition problems, under which conditions does the existence of a non-decomposable graph imply the $NP$-completeness of the decomposition problem?

EDIT For the second question, Is there a subclass of connected bridgeless cubic graphs where non-decomposable graphs exist but it is polynonial time to decide the existence of a (edge) decomposition?

The linked post on MthOverflow provides some interesting examples of connected cubic graph decomposition problems.

EDIT The problem is posted on MathOverflow: Connection between Barnette conjecture and hardness of cubic graph decomposition

• Very interesting question...care to share the polynomial time reduction from an NP-complete problem to the bicubic planar graph decomposition problem (for vertex-disjoint paths of length 2)?
– user1338
Mar 17, 2015 at 16:27
• @Philip White Here is the reduction link.springer.com/chapter/10.1007/11752578_121#page-1 Mar 17, 2015 at 16:56
• It's behind a paywall, can you just tell me what problem it's a reduction from?
– user1338
Mar 17, 2015 at 18:21
• @PhilipWhite They proved that perfect P3-matching in cubic bipartite planar 2-connected graphs is NP-complete. The reduction is from 3D matching for planar instances. Mar 17, 2015 at 18:31
• For edge decomposition what are the known results? Is it NPC to find an edge disjoint P4 decomposition on cubic bipartite graphs? (i.e. dropping the "bridgless" constraint) Mar 19, 2015 at 11:09

The idea is to find a perfect matching $M$ (which exists and can be found in polynomial time since your graph is cubic and bridgeless), and then to use the edges of $M$ as middle edges of the $P_4$'s.
To complete the construction, remove the edges of $M$ and orient the remaining cycles arbitrarily. Then attach to each edge $\{u, v\}$ of $M$ the arcs going out from $u$ and $v$, and you are done.