Counting Critical Subgraphs in k-Critical Graphs


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(n1/(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(n1/(k − 2)) for the general case k ≥ 5. We will also discuss some related problems on k-critical graphs in the final section.

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We would like to thank Asaf Shapira for providing counterexamples to some conjectures we posed in an earlier version of this paper. We are grateful to the referees for their helpful comments and suggestions which improve the presentation.

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Correspondence to Jie Ma.

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Partially supported by NSFC grant 11622110, National Key Research and Development Project SQ2020YFA070080, and Anhui Initiative in Quantum Information Technologies grant AHY150200.

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Ma, J., Yang, T. Counting Critical Subgraphs in k-Critical Graphs. Combinatorica (2021).

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Mathematics Subject Classification (2010)

  • 05C30
  • 05C35