Decomposing eulerian graphs

Definitions and mathematical statement

Let $G$ be a finite loopless undirected multigraph. It is Eulerian when it is connected and every vertex has even degree. It is six-edge-connected when deleting any set of at most five edges leaves it connected, equivalently every nonempty proper vertex subset has at least six edges to its complement. Parallel edges are counted separately in both conditions.

At every vertex $v$, prescribe a partition $\mathcal P_v$ of the incident edge set $E(v)$ into blocks of size one or two. These partitions may be chosen independently at different vertices. A cycle respects the partitions if, whenever it passes through $v$, the two incident edges it uses belong to different blocks of $\mathcal P_v$. In particular, a prescribed two-element block is a forbidden pair of consecutive edges for a cycle through that vertex.

The conjecture states that, for every six-edge-connected Eulerian $G$ and every choice of these local partitions, there exists a decomposition of $E(G)$ into cycles respecting all the partitions. A decomposition uses each edge exactly once. Each cycle is a connected two-regular subgraph, so it has no repeated vertex; two parallel edges may constitute a cycle of length two. Different cycles may share vertices.

The quantifier over prescribed partitions is universal, and the desired decomposition may depend on them. The requirement concerns the two edges of an individual cycle at a vertex, rather than forbidding different cycles from using different edges in the same prescribed block.

Short English statement

Can every six-edge-connected Eulerian graph be decomposed into cycles that avoid every prescribed local pair of edges?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3165 (OPG-141), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Decomposing eulerian graphs. Original problem and discussion. http://www.openproblemgarden.org/op/decomposing_eulerian_graphs.