Counting Critical Subgraphs in k-Critical Graphs

Abstract

Gallai asked in 1984 if any k-critical graph on n vertices contains at least n distinct (k − 1)-critical subgraphs. The answer is trivial for k ≤ 3. Improving a result of Stiebitz [10], Abbott and Zhou [1] proved in 1995 that for all k ≥ 4, any k-critical graph contains Ω(n^{1/(k − 1)}) distinct (k − 1)-critical subgraphs. Since then no progress had been made until very recently, when Hare [4] resolved the case k = 4 by showing that any 4-critical graph on n vertices contains at least (8n − 29)/3 odd cycles.

In this paper, we mainly focus on 4-critical graphs and develop some novel tools for counting cycles of specified parity. Our main result shows that any 4-critical graph on n vertices contains Ω(n²) odd cycles, which is tight up to a constant factor by infinitely many graphs. As a crucial step, we prove the same bound for 3-connected non-bipartite graphs, which may be of independent interest. Using the tools, we also give a short solution to Gallai’s problem when k = 4. Moreover, we improve the longstanding lower bound of Abbott and Zhou to Ω(n^{1/(k − 2)}) for the general case k ≥ 5. We will also discuss some related problems on k-critical graphs in the final section.