Finiteness of vertex-critical graphs for the eight remaining five-vertex forbidden graphs

Determine whether, for every integer k ≥ 1, there are finitely many k-vertex-critical (P5, H)-free graphs when H is one of the eight five-vertex graphs co-gem, chair, diamond + P1, C4 + P1, bull, P3 + 2P1, K5 − e, or K5.

Background

The paper establishes that, for every fixed integer k ≥ 1, only finitely many k-vertex-critical (P5, W4)-free graphs exist, where W4 is the four-wheel, and it completely characterizes the 5-vertex-critical graphs in this class. This resolves the corresponding finiteness question for H = W4 within the broader program of determining which five-vertex forbidden induced subgraphs H yield finite families of k-vertex-critical (P5, H)-free graphs.

The conclusion identifies eight five-vertex graphs for which the finiteness question remains unresolved: co-gem, chair, diamond + P1, C4 + P1, bull, P3 + 2P1, K5 − e, and K5. Resolving the problem requires determining finiteness for all k ≥ 1 in each of these eight hereditary graph classes. The authors note that the techniques developed for the dense graphs P3 + 2P1, K5 − e, and K5 may be especially relevant.

References

From [10] and the recent work showing finiteness in the cases where H = K1,3 + P1 and H = K3 + 2P1 [36], the only remaining open cases are when H is any of the following eight graphs of order 5:

  • co-gem
  • chair
  • diamond + P1
  • C4 + P1
  • bull
  • P3 + 2P1
  • K5 − e
  • K5
— Critical $(P_5,W_4)$-Free Graphs  (2501.04923 - Xia et al., 9 Jan 2025) in Section 7, Conclusion, p. 21

K. Cameron et al. also posed the following open question that provides the other main motivation for our results in this paper. For which graphs $H$ of order $5$ are there only finitely many $k$-vertex-critical $(P_5,H)$-free graphs for $k\ge 5$?

— A dichotomy for the number of vertex-critical ($P_5$, $H$)-free graphs when $H$ is bipartite  (2608.20045 - Beaton et al., 20 Aug 2026) in Problem 1, Section 1 (Introduction)

Instead of forbidding graphs in addition to $P_5$ incrementally by order, it may be more natural to consider forbidding graphs incrementally by chromatic number. This motivates the following open problem. For which graphs $H$ with $\chi(H)\ge 3$ is $\mathrm{crit}_{k}{P_5,H}<\infty$ for all natural numbers $k$?

— A dichotomy for the number of vertex-critical ($P_5$, $H$)-free graphs when $H$ is bipartite  (2608.20045 - Beaton et al., 20 Aug 2026) in Problem 2, Section 1 (Introduction)

From this, and computational evidence, we pose the following conjecture graphs $H$ with $\chi(H) =3$. Let $H$ be a graph with $\chi(H)=3$. Then $\mathrm{crit}_{k}{P_5,H}<\infty$ for all $k\ge 5$ if and only if $H$ is $(2P_2,K_3+P_1,C_5,\overline{C_6},\operatorname{net},\operatorname{co-net})$-free.

— A dichotomy for the number of vertex-critical ($P_5$, $H$)-free graphs when $H$ is bipartite  (2608.20045 - Beaton et al., 20 Aug 2026) in Section 5, Conclusion