Sabidussi’s compatible-circuit conjecture
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Given an Euler tour of a finite connected Eulerian multigraph of minimum degree at least four, can its edges be partitioned into connected 2-regular circuits compatible with the tour’s transitions?
Why it matters and what counts as progress
Why it matters: The compatibility condition strengthens an ordinary circuit decomposition and extends questions settled for planar and K₅-minor-free graphs.
What counts as progress: A compatible decomposition for every allowed multigraph and fixed tour, or a counterexample meeting the degree and connectivity assumptions.
How the problem developed
- 1975 · Conjecture [1]
Gert Sabidussi Posed the compatible-circuit question for Euler tours of connected Eulerian multigraphs of minimum degree at least four.
- 2000 · Partial result [2]
Jiancheng Fan and Cun-Quan Zhang Proved the conjecture for graphs with no K₅ minor.
- 14 Jul 2026 · Proof claim [3]
Mikhail Ulyanov Reported a proof for every finite connected Eulerian multigraph of minimum degree at least four and a fixed Euler tour.
AI contribution
Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture
Every finite connected Eulerian multigraph of minimum degree at least four has, for a fixed Euler tour, an edge partition into connected 2-regular circuits compatible with that tour’s transitions.
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