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.

加莱在1984年问，在n个顶点上是否有任何k临界图包含至少n个不同的（k -1）临界子图。答案是平凡的ķ ≤3.提高Stiebitz [10]，Abbott和周的结果[1]，对于所有在1995年证明ķ ≥4，任何ķ -临界图包含Ω（Ñ ^{1 /（ķ - 1）}）个不同的（k − 1）个关键子图。此后直到最近，Hare [4]通过证明n个顶点上的任何4临界图至少包含（8 ）来解决情况k = 4时，才取得进展。n -29）/ 3个奇数周期。

在本文中，我们主要关注4临界图，并开发一些新颖的工具来计算指定奇偶校验的周期。我们的主要结果表明，在n个顶点上的任何4临界图都包含Ω（n ²）个奇数周期，通过无数个图，该奇数周期都可以紧紧固定为一个常数因子。作为关键的一步，我们证明了3连通非二分图的界线相同，这可能是具有独立意义的。使用工具，当k = 4时，我们也给出了加莱问题的简短解决方案。此外，对于一般情况k，我们将Abbott和Zhou的长期下界提高到Ω（n ^{1 /（k − 2）}）。≥5。我们还将在最后一部分中讨论k临界图上的一些相关问题。