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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 containsat 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 oflength $0\bmod r$ 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 $1\bmod k$ amongall $(k+1)$-critical graphs, strengthening a recent result of Moore and Westand extending it to the $k=3$ case.

4-Chromatic Graphs Have At Least 4 Cycles of Length $0 \bmod 3$