Cayley Graph C(36; ±{1, 4, 10, 17}) Removes 5-Team Clusters of 9 Matches in UEFA League Phase

UEFA replaced the 32-team group stage with a 36-team league phase in 2024-25. Each side plays eight opponents drawn two from each pot. The resulting 144-edge graph on 36 vertices produces repeated dense 5-team subgraphs. UCL 2024-25 and UEL 2024-25 each contained a 5-team set with nine internal matches; UCL 2025-26 repeated the pattern.

Data from five completed or scheduled seasons show 225 to 332 five-team subsets with at least seven internal matches and three to five subsets with nine matches. The UECL, limited to six matches per team, records zero such clusters. Random draws therefore reliably generate local round-robin concentrations that reduce collective advancement probability for the affected clubs.

The circulant Cayley graph C(36; ±{1,4,10,17}) is 8-regular, vertex-transitive, and contains no K5. Replacing the current pot-based draw with this fixed connection set eliminates all 5-team clusters of eight or more matches. Only the country protection rule requires adjustment in UCL and UEL; UECL needs none.

UEFA can adopt the graph for the 2026-27 season by publishing the 36-vertex adjacency list before the pot draw. Verification requires one line of code against the published schedule.