We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $k\in\{3,4,5\}$, and by Fraigniaud et al. [PODC 2024] for $k\geq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what is done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the ''local density'' of graphs without $2k$-cycles, which we believe is interesting on its own.

翻译：我们证明，对于任意 $k\geq 2$，在广播CONGEST模型中，$C_{2k}$-无环性可以通过确定性算法在 $O(n^{1-1/k})$ 轮内判定。对于 $k=2$ 的情况，由于Drucker等人[PODC 2014]针对 $C_4$-无环性给出的下界（该下界甚至适用于随机算法），该（确定性）轮复杂度在忽略对数因子后是最优的。此外，该结果与Censor-Hillel等人[DISC 2020]针对 $k\in\{3,4,5\}$ 以及Fraigniaud等人[PODC 2024]针对 $k\geq 6$ 提出的最佳已知随机算法的轮复杂度相匹配。我们的算法采用并行广度优先搜索探索，并在每轮中确定性选择待转发的路径集合，其方式类似于Korhonen和Rybicki[OPODIS 2017]用于检测奇数长度环的方法。然而，我们算法设计与分析的关键要素是一个新的组合学结论，该结论限定了不含 $2k$-环的图的“局部密度”，我们认为这一结论本身具有独立的研究价值。