A dichotomy for the number of vertex-critical ($P_5$, $H$)-free graphs when $H$ is bipartite

Abstract: A graph $G$ is $k$-vertex-critical if $χ(G)=k$, but $χ(H)<k$ for every induced subgraph $H$ of $G$. A graph $G$ is $(H_1,H_2,\dots,H_m)$-free if does not contain $H_i$ as an induced subgraph for any $i\in{1,2,\dots,m}$.We provide the following dichotomy that for bipartite graphs $H$ and any fixed integer $k\ge 5$ , there are only finitely many $k$-vertex-critical $(P_5,H)$-free graphs if and only if $H$ is $2P_2$-free. This leads us to pose the problem about determining for which graphs $H$ with $χ(H)\ge 3$ there are infinitely many $k$-vertex-critical $(P_5,H)$-free graphs for all $k\ge 5$. Toward this problem, we show that there only finitely many $k$-vertex-critical $(P_5, K_{s,t}+e)$-free graphs for all $k,s,t\ge 1$, where $K_{s,t}+e$ is a complete bipartite graph plus a single edge. On the other hand, we show that there are infinitely many $k$-vertex-critical $$(P_5,\operatorname{net},\operatorname{co-net},\overline{C_5},\overline{C_6},\dots\overline{C_{k-1}})$$-free graphs for all $k\ge 5$. We also show that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,\overline{L(K_{2,n})})$-free graphs for all $\ell,n\ge 0$, providing the largest known subfamily of $(P_4+\ell P_1)$-free graphs to satisfy this property. Our results, together with known results, imply the existence of new polynomial-time certifying algorithms to determine the $k$-colourability of many subfamilies of $P_5$-free and $(P_4+\ell P_1)$-free graphs for fixed $k\ge 5$. Our proof techniques apply a powerful theorem of Chudnovsky, Kim, Oum, and Seymour (2016) on prime graphs that we expect to be of interest and have further applications to bounding the number of $k$-vertex-critical graphs in other hereditary families of graphs.

• The paper proves that for every k ≥ 5 and bipartite H, only finitely many k-vertex-critical (P₅,H)-free graphs exist exactly when H is 2P₂-free, using prime-graph reductions and unavoidable induced subgraphs.
• The paper resolves the outstanding cases H = P₄ + P₁ and H = C₄ + P₁, extends finiteness to forbidden graphs of the form Kₛ,ₜ + e, and establishes new finite and infinite families for non-bipartite obstructions.
• The results support polynomial-time certifying k-Colouring algorithms in the finite cases while leaving major questions open for higher-chromatic H, including K₅ − e, K₅, and the Cameron–Hoàng–Sawada conjecture.

This paper by Beaton and Cameron (2608.20045) establishes a complete finiteness dichotomy for $k$-vertex-critical $(P_5,H)$-free graphs when $H$ is bipartite, and contributes several new positive and negative results toward two long-standing open problems on vertex-critical graphs in hereditary classes. The central tool is the theorem of Chudnovsky, Kim, Oum, and Seymour on unavoidable induced subgraphs in large prime graphs.

Background and motivation

A graph is $k$-vertex-critical if $\chi(G)=k$ but every proper induced subgraph has chromatic number less than $k$. If a hereditary class contains only finitely many $(k+1)$-vertex-critical graphs, then $k$-Colouring restricted to that class admits a polynomial-time certifying algorithm: a "yes" certificate is a colouring and a "no" certificate is an induced $(k+1)$-vertex-critical subgraph. This motivates determining for which forbidden induced subgraphs $H$ the quantity $(P_5,H)$0 — the number of $(P_5,H)$1-vertex-critical $(P_5,H)$2-free graphs — is finite.

Two problems drive the paper. The first is the conjecture of Cameron, Hoàng, and Sawada that for $(P_5,H)$3, there are finitely many $(P_5,H)$4-vertex-critical $(P_5,H)$5-free graphs if and only if $(P_5,H)$6 is an induced subgraph of $(P_5,H)$7 for some $(P_5,H)$8. The second is the question of K. Cameron, Goedgebeur, Huang, and Shi asking for which graphs $(P_5,H)$9 of order 5 there are finitely many $H$0-vertex-critical $H$1-free graphs for all $H$2. Prior to this work, five graphs of order 5 remained unresolved: $H$3, $H$4, $H$5, $H$6, and $H$7.

The prime graph framework

The paper's main proof technique rests on a reduction to prime graphs. The authors prove (via induction on $H$8, using the fact that homogeneous sets in a $H$9-vertex-critical graph induce $k$0-vertex-critical graphs for $k$1) that if the order of every $k$2-colourable prime graph in a hereditary class is bounded, then the class contains only finitely many $k$3-vertex-critical graphs. Combined with the unavoidable-subgraph theorem of Chudnovsky et al., this yields a corollary: for any $k$4 and $k$5, only finitely many prime graphs avoid all of $k$6, $k$7, $k$8, $k$9, and $\chi(G)=k$0. A short lemma shows that any chain of length at least $\chi(G)=k$1 contains an induced $\chi(G)=k$2 or an induced $\chi(G)=k$3, which eliminates the "chain" case from the unavoidable list.

The bipartite dichotomy

The key structural observation is that $\chi(G)=k$4, $\chi(G)=k$5, and $\chi(G)=k$6 each contain an induced $\chi(G)=k$7, while the half-graph $\chi(G)=k$8 contains induced copies of both $\chi(G)=k$9 and $k$0. Consequently:

• For all $k$1, there are finitely many $k$2-vertex-critical $k$3-free graphs.
• For all $k$4, there are finitely many $k$5-vertex-critical $k$6-free and $k$7-free graphs. The cases $k$8 resolve two of the five outstanding instances of the order-5 problem, namely $k$9 and $(k+1)$0.

The main theorem then follows from the fact that $(k+1)$1 is universal for $(k+1)$2-free bipartite graphs (Lozin and Rudolf): for bipartite $(k+1)$3 and any $(k+1)$4, there are finitely many $(k+1)$5-vertex-critical $(k+1)$6-free graphs if and only if $(k+1)$7 is $(k+1)$8-free. The necessity direction uses the known infinitude of $(k+1)$9-vertex-critical $k$0-free graphs for $k$1; the sufficiency direction embeds any $k$2-free bipartite $k$3 into a sufficiently large half-graph via a nested-neighbourhood argument. This is the largest subfamily of $k$4-free graphs for which finiteness of $k$5-vertex-critical graphs has been established, and it generalizes prior results for $k$6 being complete bipartite, banner, $k$7, $k$8, or chair.

Non-bipartite $k$9: finiteness and infinitude

Toward the natural refinement — for which $(k+1)$0 with $(k+1)$1 is $(k+1)$2 finite for all $(k+1)$3? — the paper proves two complementary types of results.

On the positive side, for all $(k+1)$4 and $(k+1)$5, there are finitely many $(k+1)$6-vertex-critical $(k+1)$7-free graphs, where $(k+1)$8 is a complete bipartite graph plus one edge within one part. The proof assumes a large induced half-graph exists in a critical graph, uses Lemma on anticomplete pairs to extract vertices with controlled adjacency to the half-graph, applies a Ramsey argument $(k+1)$9 to obtain a stable set of size $H$0, and derives an induced $H$1 — a contradiction. This generalizes earlier finiteness results for $H$2, $H$3, and cricket.

On the negative side, the authors analyze the circulant-style graphs $H$4 of Cameron and Hoàng, which are $H$5-vertex-critical, $H$6-free, and $H$7-free. New structural lemmas show that the smallest hole in $H$8 has length at least $H$9 (each residue class modulo $(P_5,H)$00 must appear on any cycle of length at least 4 in the complement), and that $(P_5,H)$01 is net-free and co-net-free. Hence for all $(P_5,H)$02 there are infinitely many $(P_5,H)$03-vertex-critical $(P_5,H)$04-free graphs when $(P_5,H)$05 is net, co-net, or $(P_5,H)$06 for $(P_5,H)$07. Via the Strong Perfect Graph Theorem, the last result implies that any $(P_5,H)$08 satisfying the chromatic-number-refined problem must be perfect.

For the $(P_5,H)$09-free setting, the same prime-graph framework yields finiteness of $(P_5,H)$10-vertex-critical $(P_5,H)$11-free graphs for all $(P_5,H)$12 — the largest known such subfamily, and further evidence for the Cameron–Hoàng–Sawada conjecture, which remains open even for $(P_5,H)$13, $(P_5,H)$14.

Computational results for $(P_5,H)$15-containing obstructions

Since any $(P_5,H)$16 satisfying the refined problem must contain a triangle but no $(P_5,H)$17 (as $(P_5,H)$18 for $(P_5,H)$19 while finite for $(P_5,H)$20), the authors examine the four $(P_5,H)$21-free order-6 graphs containing an induced $(P_5,H)$22. They show $(P_5,H)$23 is $(P_5,H)$24-free and $(P_5,H)$25-free, so these do not obstruct finiteness. Exhaustive generation using Jooken's program gives exact counts:

Forbidden $(P_5,H)$26	Number of 5-vertex-critical $(P_5,H)$27-free graphs
twin-$(P_5,H)$28	287
$(P_5,H)$29	188

These lists are publicly available in graph6 format. Based on this evidence, the authors conjecture that for $(P_5,H)$30, finiteness holds exactly when $(P_5,H)$31 is $(P_5,H)$32-free.

Limitations and open questions

Several questions remain explicitly open. The remaining three cases of the order-5 problem ($(P_5,H)$33, $(P_5,H)$34, $(P_5,H)$35) all have chromatic number at least 4, and the only 4-chromatic $(P_5,H)$36 for which finiteness is known is $(P_5,H)$37. The Cameron–Hoàng–Sawada conjecture is unresolved even in its most restrictive nontrivial instance. It is also open whether $(P_5,H)$38 is finite when $(P_5,H)$39 contains an induced $(P_5,H)$40. The finiteness results here establish existence of polynomial-time certifying algorithms but do not provide explicit bounds on the number or size of critical graphs beyond finiteness itself, and the computational counts cover only $(P_5,H)$41.

Conclusion

The paper delivers a clean dichotomy for bipartite $(P_5,H)$42 — finiteness holds precisely when $(P_5,H)$43 is $(P_5,H)$44-free — resolves two outstanding cases of the order-5 problem, extends finiteness to all graphs $(P_5,H)$45, and demonstrates infinitude for net, co-net, and complements of short cycles. Methodologically, it demonstrates that building arguments around arbitrarily large induced half-graphs (and, analogously, $(P_5,H)$46) is a viable template for bounding vertex-critical graphs in other hereditary classes, complementing earlier techniques based on large induced $(P_5,H)$47.