欧足联的新欧冠联赛32进36队单循环赛制存在设计缺陷，导致某些球队形成不公平的密集分组集群1。在2025-26赛季的首批抽签中，那不勒斯、本菲卡、切尔西、阿贾克斯、卡拉巴赫5支球队的10场对阵中有9场将在小组赛中进行1。类似现象在2024-25赛季也已出现，勒沃库森、马竞、费耶诺德、红牛萨尔茨堡、布拉格斯巴达这5队同样在9场对阵中密集相遇1。
密集集群的出现显著降低了相关球队的晋级概率1。模拟数据表明，若5支等实力球队处于集群中，全部晋级的概率将从17%跌至10%；按实际等级分计算，那不勒斯等5队全部晋级的概率仅为1.4%，相比同等级球队的4.2%下降了68%1。
研究提出了基于凯莱图的解决方案1。采用凯莱图C(36; ±{1, 4, 10, 17})作为配对结构，可在数学上保证每支球队所处位置完全对称公平，任何5队集群内部对阵不超过6场（相比当前最多9场），且消除三角形结构1。该方案需修改欧足联的同国对阵规则，具体可将一支球队面对同国对手的上限从2提升至3，或将大联赛分为子组1。技术上该方案可在直播抽签仪式中实现，每抽一球约需55毫秒计算时间，整个36球抽签过程耗时约2秒1。值得注意的是，欧洲联盟杯（UECL）无需规则修改即可部署该方案1。
UEFA's redesigned Champions League format for the 2025-26 season, which expands to 36 teams competing in a single round-robin stage, contains a structural flaw that produces mathematically unfair group clusters.1 In the initial draw, five teams—Napoli, Benfica, Chelsea, Ajax, and Karabakh—were scheduled to face each other in nine matches, with nine of these fixtures occurring during the group stage alone.1 Analysis of the 2024-25 season, the format's inaugural year, revealed a similar problem: Bayer Leverkusen, Atlético Madrid, Feyenoord, RB Salzburg, and Sparta Prague were assigned nine mutual matchups.1
These dense clusters place affected teams at a competitive disadvantage. Simulation data shows that five equally matched teams within such a cluster have only a 10% probability of all advancing, compared to 17% for teams in non-clustered configurations.1 Using actual UEFA rankings, the five-team cluster containing Napoli faces merely a 1.4% chance of complete advancement, a 68% decline relative to the 4.2% probability for equivalently ranked teams outside clusters.1
A Cayley graph mathematical structure offers a remedy. The proposed model, C(36; ±{1, 4, 10, 17}), would ensure complete symmetry across all team positions and limit intra-cluster matches to a maximum of six, compared to the current nine, while eliminating triangle-shaped matchup configurations.1 Implementation requires modifying UEFA's rule restricting same-nation opponents—either increasing the ceiling from two to three matches or dividing top leagues into subgroups.1 The Europa Conference League requires no such adjustment.1 The solution is computationally feasible for live draws, requiring approximately 55 milliseconds per team assignment, or roughly two seconds for all 36 teams.1
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