UEFA's Champions League draw creates unfair clusters; a Cayley graph fixes it
A technical analysis suggests that the new UEFA Champions League format creates unintended clusters of teams playing each other disproportionately often. The author proposes using a symmetric Cayley graph to optimize the draw and ensure a more balanced competition.
Why it matters
This highlights the intersection of sports management and graph theory, demonstrating how mathematical modeling can improve the fairness of large-scale tournament structures.
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.
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