4-Chromatic Graphs Have At Least 4 Cycles of Length
S Kim, ME Picollelli - arXiv preprint arXiv:2312.05945, 2023 - arxiv.org
S Kim, ME Picollelli
arXiv preprint arXiv:2312.05945, 2023•arxiv.orgA 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 …
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 …
A 2018 conjecture of Brewster, McGuinness, Moore, and Noel asserts that for , if a graph has chromatic number greater than , then it contains at least as many cycles of length as the complete graph on vertices. Our main result confirms this in the case by showing every -critical graph contains at least cycles of length , and that is the unique such graph achieving the minimum. We make progress on the general conjecture as well, showing that -critical graphs with minimum degree have at least as many cycles of length as , provided . We also show that uniquely minimizes the number of cycles of length among all -critical graphs, strengthening a recent result of Moore and West and extending it to the case.
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