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UEFA's Champions League draw creates unfair clusters; a Cayley graph fixes it

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Why This Matters

This article highlights how UEFA's new Champions League format can inadvertently create tightly knit clusters of teams that play each other excessively, reducing fairness and competitive diversity. Implementing a symmetric Cayley graph structure offers a provably fair solution, improving tournament integrity and ensuring more balanced matchups for both players and fans.

Key Takeaways

A tight little cluster hidden in the 2025-26 Champions League draw shows why the “new format” needs one small tweak, and gives a beautiful excuse to talk about symmetric graphs.

TL;DR: UEFA’s new league-phase format can accidentally produce small clusters of teams that play each other disproportionately often, hurting their collective chance of advancing all at once. A single symmetric graph on 36 vertices, C(36; ±{1, 4, 10, 17}), eliminates this risk provably. Deploying it needs one small tweak to the country rule for UCL/UEL, and no tweak at all for UECL.

Napoli, Benfica, Chelsea, Ajax, Qarabağ.

In the 2025-26 UEFA Champions League, they are scheduled to play each other in nine of the ten possible pairings. Only Napoli-Ajax is missing from a complete round-robin among the five. The other nine (Napoli-Benfica, Benfica-Chelsea, Chelsea-Ajax, Ajax-Qarabağ, and five more) all happen in the league phase.

UCL 2025-26 - 9 of 10 possible games scheduled.

That’s an oddly tight little cluster in a 36-team competition. And it’s the kind of thing that isn’t supposed to happen anymore.

Except it does, every season, in almost every UEFA continental competition. In UCL 2024-25 (the first year of the new format), Bayer Leverkusen, Atlético Madrid, Feyenoord, RB Salzburg and Sparta Prague play each other 9 of 10 times. In UEL 2024-25, Roma, Tottenham, AZ, Galatasaray, and IF Elfsborg do the same. UEL 2025-26 has a maximum 5-team cluster of 8 out of 10; UECL 2025-26 has 7 out of 10.

UCL 2024-25 — 9 of 10 UEL 2024-25 — 9 of 10 UEL 2025-26 — 8 of 10 UECL 2025-26 — 7 of 10

Click to expand: full stats across all five competitions **How many dense clusters, per season?** Season 5-team clusters with ≥7 matches ≥8 matches =9 matches 4-team clusters with ≥5 matches =6 matches UCL 2024-25 225 17 3 84 4 UCL 2025-26 332 29 1 109 5 UEL 2024-25 240 17 2 87 3 UEL 2025-26 146 6 0 60 0 UECL 2025-26 (6 matches, not 8) 35 0 0 14 0 Two things jump out: (i) every completed UCL/UEL season has produced a cluster with 8 or more internal matches, and (ii) 2025-26 UCL is actually the season with the most 8+ clusters we have on record (29 subsets), though only one hits 9. UECL is meaningfully less dense because it uses a sparser structure (6 opponents per team, not 8).

What UEFA changed in 2024

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