// HACKER NEWS — CYBERSECURITY
UEFA's Champions League draw creates unfair clusters; a Cayley graph fixes it
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.
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.
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.
For decades, the Champions League and its sister competitions used the same shape: 32 teams, 8 groups of 4, a round-robin within each group. Each team played 6 group-stage matches (three opponents faced twice each) before the top two per group advanced.
In 2024-25, UEFA scrapped that. In its place: a single 36-team “league phase” where each team plays 8 different opponents (2 from each of 4 seeded pots), and a single 36-team table sorts everyone into three buckets: top 8 auto-advance to the round of 16, teams ranked 9-24 go to a knockout playoff, teams 25-36 are eliminated. The same format runs across all three UEFA men’s competitions: Champions League (UCL), Europa League (UEL), Conference League (UECL).
The sales pitch was reasonable. More marquee matches. More competitive standings. No more “dead rubber” final matchday when both advancing teams have qualified already. And the “single table” framing removes the arbitrary group draw, which had famously produced imbalanced “groups of death” for years.
But it introduced a subtler risk that nobody flagged at the time.
Here’s the thing about 36 teams each playing 8 opponents: it produces a huge network of 144 matches, and inside that network you can find small clusters of teams that happen to play each other disproportionately often. Not because anyone designed it that way; because the draw is random and 144 edges among 36 teams have to land somewhere.