A 2018 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for $k \ge 3$, if a graph has chromatic number greater than $k$, then it contains at least as many cycles of length $0 \bmod k$ as the complete graph on $k+1$ vertices. Our main result confirms this in the $k=3$ case by showing every $4$-critical graph contains at least $4$ cycles of length $0 \bmod 3$, and that $K_4$ is the unique such graph achieving the minimum. We make progress on the general conjecture as well, showing that $(k+1)$-critical graphs with minimum degree $k$ have at least as many cycles of length $0\bmod r$ as $K_{k+1}$, provided $k+1 \ne 0 \bmod r$. We also show that $K_{k+1}$ uniquely minimizes the number of cycles of length $1\bmod k$ among all $(k+1)$-critical graphs, strengthening a recent result of Moore and West and extending it to the $k=3$ case.

翻译：Brewster、McGuinness、Moore和Noel在2018年提出猜想：对于$k \ge 3$，若一个图的色数大于$k$，则它包含的长度为$0 \bmod k$的环的数量至少与$k+1$个顶点的完全图一样多。我们的主要结果在$k=3$情形下证实了该猜想，证明了每个$4$-临界图至少包含4个长度为$0 \bmod 3$的环，且$K_4$是达到该最小值的唯一此类图。我们也在一般猜想上取得了进展，证明了当$k+1 \ne 0 \bmod r$时，最小度为$k$的$(k+1)$-临界图包含的长度为$0 \bmod r$的环的数量至少与$K_{k+1}$一样多。我们还证明了$K_{k+1}$在所有$(k+1)$-临界图中唯一地最小化了长度为$1 \bmod k$的环的数量，这加强了Moore和West的最新结果，并将其推广到$k=3$情形。