Vertex-critical co-gem-free graphs

Abstract: A graph $G$ is $k$-$colorable$ if $V(G)$ can be partitioned into at most $k$ stable sets. A graph $G$ is $k$-$chromatic$ if $k$ is the smallest integer for which $G$ is $k$-colorable. In general, for a fixed $k\ge 3$, determining whether an arbitrary graph $G$ is $k$-colorable is NP-complete. Consequently, $k$-coloring algorithms for restricted graph classes, such as $\mathcal{H}$-free graphs, have been widely studied over the past few decades. A graph $G$ is $k$-$vertex$-$critical$ if $G$ is $k$-chromatic and every proper induced subgraph of $G$ is ($k$-1)-colorable. Given a graph $G$, most of the certifying $k$-coloring algorithms in the literature either output a $k$-coloring of $G$ or a ($k$+1)-vertex-critical induced subgraph of $G$, thus, proving that $G$ is not $k$-colorable. As a result, $k$-vertex-critical graphs have gathered considerable attention in the recent years. Beaton and Cameron [Vertex-critical graphs in co-gem-free graphs, Theoretical Computer Science 1042 (2025) 115234] asked for which graphs $H$ of order five are there finitely many $k$-vertex-critical (co-gem, $H$)-free graphs for all $k$? In this paper we explore the structure of (co-gem, house)-free graphs and (co-gem, dart)-free graphs, and prove that, for each $k\ge 1$, there are finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, when $H$ is in ${$house, dart$}$.

• The paper proves that, for every fixed k, only finitely many k-vertex-critical graphs are both co-gem-free and house-free or dart-free, resolving two cases of the Finiteness Problem.
• The authors combine prime-graph decompositions, homogeneous-set reductions, clique-cutset arguments, Ramsey bounds, and layered neighborhood counting to establish bounded-size critical obstructions.
• These finiteness results yield polynomial-time certifying k-coloring algorithms, although explicit obstruction lists and tight bounds remain open computational goals.

This paper by Belavadi and Karthick resolves two open cases of the Finiteness Problem for vertex-critical graphs in co-gem-free hereditary classes. Specifically, it proves that for every $k \ge 1$ there are only finitely many $k$-vertex-critical (co-gem, house)-free graphs and finitely many $k$-vertex-critical (co-gem, dart)-free graphs, answering a question posed by Beaton and Cameron (2606.11757).

Background and motivation

A graph is $k$-vertex-critical if it is $k$-chromatic and every proper induced subgraph is $(k-1)$-colorable. These graphs matter algorithmically: if, for fixed $k$, a hereditary class $\mathcal{G}$ contains only finitely many $k$-vertex-critical graphs, then $k$-colorability of graphs in $k$0 can be decided in polynomial time by a certifying algorithm that either outputs a $k$1-coloring or exhibits a $k$2-vertex-critical induced obstruction. This folklore reduction motivates what Huang and Li termed the Finiteness Problem: given a hereditary class $k$3 and integer $k$4, are there finitely many $k$5-vertex-critical graphs in $k$6?

The landscape of known results frames the contribution. Chudnovsky, Goedgebeur, Schaudt, and Zhong settled the problem completely for $k$7-free graphs with $k$8: finitely many 4-vertex-critical $k$9-free graphs exist exactly when $k$0 is an induced subgraph of $k$1, $k$2, or $k$3. For $k$4, Hoàng et al. showed there are infinitely many $k$5-vertex-critical $k$6-free graphs, so attention shifted to $k$7-free classes and to related families such as co-gem-free graphs. Beaton and Cameron proved finiteness for (co-gem, $k$8)-free graphs when $k$9 has four vertices, or when $k$0, and asked which other five-vertex graphs $k$1 yield finiteness. The present paper answers this for $k$2.

Finiteness for (co-gem, house)-free graphs

The proof strategy combines three ingredients. First, a structural theorem of Brandstädt and Kratsch states that a prime (co-gem, house)-free graph is either one of finitely many specific graphs on at most nine vertices, perfect, or admits a partition into a $k$3, a clique $k$4 complete to it, and a $k$5-free set $k$6 with uniform neighborhoods on edges of the cycle. Second, a lemma of Xia–Jooken–Goedgebeur–Huang shows that components of homogeneous sets in a $k$7-vertex-critical graph are themselves vertex-critical of smaller chromatic number. Third, a general "skeleton" theorem of Belavadi and Hoàng reduces bounding all $k$8-vertex-critical graphs in a class to bounding $k$9-vertex-critical blowups of prime graphs.

The argument proceeds by induction on $k$0. If the prime base graph is perfect, its blowup is perfect and hence has exactly $k$1 vertices. If it is one of the nine-vertex exceptional graphs, any blowup has at most $k$2 vertices since each vertex is replaced by a clique of size at most $k$3. In the remaining case, the pair $k$4 forms a clique cutset, which persists under clique substitution; but Dirac's classical result that no $k$5-vertex-critical graph contains a clique cutset rules this case out entirely. Consequently the size of every $k$6-vertex-critical (co-gem, house)-free graph is bounded by a constant, and the polynomial-time certifying $k$7-coloring algorithm follows immediately.

Structure of prime (co-gem, dart)-free graphs

For the dart-free case, the paper first establishes that every prime (co-gem, dart)-free graph excludes a particular six-vertex graph $k$8: assuming an induced $k$9, primality forces a vertex distinguishing a non-homogeneous pair, and a short case analysis shows each possible adjacency pattern creates either a dart or a co-gem.

When such a prime graph contains a $(k-1)$0 induced by $(k-1)$1, the neighborhood of $(k-1)$2 partitions into sets $(k-1)$3 (vertices seeing $(k-1)$4, optionally also $(k-1)$5), cliques $(k-1)$6, $(k-1)$7, $(k-1)$8 classified by their adjacency to $(k-1)$9, and $k$0 (vertices complete to $k$1). Six structural properties are derived from dart/co-gem avoidance: each $k$2 is a stable set or a clique; the $k$3, $k$4, $k$5 are cliques; $k$6 is anticomplete to $k$7 and forces $k$8; nonempty $k$9 forces each $\mathcal{G}$0 to be a clique; nonadjacent pairs in $\mathcal{G}$1 dominate $\mathcal{G}$2; and stable $\mathcal{G}$3 sets are complete to neighboring $\mathcal{G}$4's and anticomplete to the others.

Two further lemmas drive the finiteness proof. A modified version of a lemma from Xia et al. shows that in a dart-free graph, if $\mathcal{G}$5 is bounded, not complete to a co-connected set $\mathcal{G}$6, every anticomplete pair in $\mathcal{G}$7 is hit by some vertex of $\mathcal{G}$8, and a vertex outside $\mathcal{G}$9 is complete to $k$0, then $k$1 is bounded by a function of $k$2 — via a layered expansion argument where $k$3 and the layers terminate within $k$4 steps. Combining these, the paper proves that for each $k$5 there are finitely many prime $k$6-colorable (co-gem, dart)-free graphs containing a $k$7: when $k$8, the pigeonhole principle bounds each stable $k$9 by $k$0, giving $k$1; when $k$2, the layered lemma bounds each anti-component of $k$3.

An independent structural result

The paper also proves a lemma stated as being of independent interest: every co-connected (co-gem, dart, $k$4)-free graph is either perfect or $k$5-free. The proof works in the complement, showing that connected (gem, co-dart, $k$6)-free graphs containing a smallest odd hole $k$7 ($k$8) are triangle-free. Four claims establish that common neighborhoods of adjacent cycle vertices are empty, exterior vertices attach to $k$9 in a tightly constrained pattern, $k$00 is triangle-free, and finally — by induction over distance layers from $k$01 — that every neighborhood in the entire graph is a stable set, which precludes triangles in a connected graph.

Finiteness for (co-gem, dart)-free graphs

The main dart-free theorem follows by combining the pieces. For a prime (co-gem, dart)-free base graph $k$02: if $k$03 is perfect, its only $k$04-vertex-critical blowup is $k$05. If $k$06 is imperfect but $k$07-free, the structural lemma forces $k$08 to be $k$09-free, and blowups of $k$10-free graphs remain $k$11-free, so any $k$12-vertex-critical blowup has at most $k$13 vertices by Ramsey's theorem. If $k$14 contains a $k$15, then $k$16 itself must be $k$17-colorable whenever a blowup is $k$18-vertex-critical, so the boundedness of prime $k$19-colorable graphs containing a $k$20 bounds $k$21, and hence $k$22. As with the house-free case, this yields a polynomial-time certifying algorithm for $k$23-colorability of (co-gem, dart)-free graphs for every fixed $k$24.

Limitations and open questions

The results are existential rather than explicit: the proofs bound the number of vertices of critical graphs but do not enumerate them, and the constants involved (such as $k$25 and $k$26) are far from tight. Goedgebeur and Schaudt's exhaustive generation algorithm could in principle produce explicit lists for small $k$27, but termination is not guaranteed a priori. The broader question of Beaton and Cameron remains open for other five-vertex graphs $k$28 beyond house and dart, and for $k$29-free graphs with $k$30 the sole unresolved case is $k$31 with $k$32. Whether the certifying algorithms obtained here can be made practical depends on explicitly determining the finite obstruction sets, which this paper does not undertake.

Conclusion

The paper settles the Finiteness Problem affirmatively for (co-gem, house)-free and (co-gem, dart)-free graphs for all $k$33, extending the program initiated by Beaton and Cameron on five-vertex obstructions in co-gem-free classes. Methodologically, it demonstrates how structural decompositions of prime graphs, clique-cutset arguments inherited from Dirac, Ramsey bounds for $k$34-free graphs, and layered counting arguments around a $k$35 skeleton combine into a reusable template for proving finiteness. Each result directly yields a polynomial-time certifying $k$36-coloring algorithm for the corresponding class.