UEFA’s 36-team league phase was designed to replace the old group-stage lottery with a single-table competition. But a graph-analysis proposal argues that the new format has created a different kind of draw inequity: small groups of clubs can be scheduled against one another far more often than expected.
The suggested remedy is not another randomization rule. It is a fixed, symmetric match network built using a mathematical structure called a Cayley graph.
The issue: concentrated schedules inside a single table
In the Champions League league phase, each club plays eight different opponents. That creates a network of 36 teams and 144 matches. A random draw can place an unusually high share of those matches within a small set of teams.
The 2025-26 draw, according to the analysis, includes Napoli, Benfica, Chelsea, Ajax and Qarabağ in nine of their 10 possible head-to-head pairings. Similar five-team clusters appeared in the 2024-25 Champions League and Europa League draws.
That concentration matters because matches within the group redistribute points among the same teams rather than allowing them to earn points against the rest of the field. It also links their outcomes: one team’s win is directly another cluster member’s loss. For a group of clubs trying to clear the top-24 cutoff together, a dense internal schedule makes their collective path harder.
The author’s simulations estimate that, for five equally strong clubs, the chance that all five reach the top 24 falls as the number of internal fixtures rises. The conclusion is not that any individual club is doomed by a cluster, but that the draw can assign meaningfully different structural positions to teams before a ball is kicked.
Why the current constraint set can amplify the problem
UEFA already prevents same-country league-phase matchups. That means clubs from countries with multiple entrants cannot all land in the same dense cluster. Teams from smaller associations, with fewer compatriots protected by that rule, can face more of the clustering exposure.
This is a useful systems-design lesson beyond football. Random assignment is often treated as synonymous with fair assignment. It is not. When participants occupy different positions in an underlying network, randomness can distribute materially unequal exposure to risk—even when the rules are applied correctly.
The proposed fix: make the network symmetric first
The proposal uses a 36-vertex Cayley graph, written as `C(36; ±{1, 4, 10, 17})`, as the pre-defined pattern of who can play whom. In practical terms, every club would be assigned to one of 36 positions, and the graph would specify eight opponent positions for each position.
The important property is symmetry. Rather than hoping a randomized draw avoids dense local clusters, the schedule structure is designed so no assigned position has a systematically worse neighborhood than another. The source argues that this eliminates the relevant clustering risk by construction.
Clubs could still be randomly assigned to graph positions, preserving uncertainty in the draw. But the randomized step would operate over positions with equivalent structural properties, rather than generate the entire match network from scratch.
The operational trade-off
A fixed graph is only useful if it works alongside UEFA’s commercial and regulatory constraints—especially seeded pots and restrictions on clubs from the same country. The proposal says deployment in the Champions League and Europa League would require a small adjustment to the country rule, while the Conference League could use the approach without such a change.
That is the real implementation question. Scheduling algorithms in sport, aviation, marketplaces and workforce systems are rarely optimized for one objective. They balance fairness, variety, travel, broadcast value, compliance and explainability. A mathematically elegant network still needs an operationally acceptable assignment process.
What to watch next
The immediate opportunity is not necessarily for UEFA to adopt this specific graph. It is to publish structural fairness metrics for each draw: cluster density, opponent-overlap measures and the distribution of schedule positions across clubs.
For competition operators, the broader takeaway is straightforward: test randomized systems not only for rule compliance, but for the distribution of opportunities they create. If a recurring bad outcome can be ruled out by design, that is generally better governance than treating it as bad luck.
