4-Chromatic Graphs Have At Least Four Cycles of Length 0 3

A 2016 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for k ≥ 3, if a graph has chromatic number greater than k, then it contains at least as many cycles of length 0 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 four cycles of length 0 3, and that K₄ 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 r as Kₖ₊₁, provided k+1 ≠ 0 r. We also show that Kₖ₊₁ uniquely minimizes the number of cycles of length 1 k among all $(k+1)$-critical graphs, strengthening a recent result of Moore and West and extending it to the $k=3$ case.