# Counting critical subgraphs in $k$-critical graphs

## TL;DR

This paper investigates the number of critical subgraphs in $k$-critical graphs, providing new bounds especially for 4-critical graphs by developing tools to count cycles of specific parity, and extends results to $k extgreater 4$.

## Contribution

The paper introduces novel methods for counting cycles of specified parity in 4-critical graphs and improves lower bounds for the number of critical subgraphs in $k$-critical graphs for $k extgreater 4.

## Key findings

- Any 4-critical graph on n vertices contains Ω(n^2) odd cycles.
- The same bound applies to 3-connected non-bipartite graphs.
- Improved lower bound of Ω(n^{1/(k-2)}) for k-critical graphs with k ≥ 5.

## 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 $k\leq 3$. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that for all $k\geq 4$, such graph contains $\Omega(n^{1/(k-1)})$ distinct $(k-1)$-critical subgraphs. Since then no progress had been made until very recently, Hare 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 $\Omega(n^2)$ odd cycles, which is tight up to a constant factor by infinite 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 very short proof for the case $k=4$. Moreover, we improve the longstanding lower bound of Abbott and Zhou to $\Omega(n^{1/(k-2)})$ for the general case $k\geq 5$. We will also discuss some related problems on $k$-critical graphs in the final section.