Vertex-critical graphs in subfamilies of $(P_4+\ell P_1)$-free graphs

Abstract: A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)&lt;k$ for all $v\in V(G)$. In this paper we make progress on the open problem of the finiteness of $k$-vertex-critical $(P_4+\ell P_1)$-free graphs by showing that there are only finitely many $k$-vertex-critical graphs in the following subfamilies of $(P_4+\ell P_1)$-free graphs for all $k\ge 1$ and $\ell\ge 0$: $\bullet$ $(P_4+\ell P_1,\text{chair})$-free graphs, $\bullet$ $(P_4+\ell P_1,P_5,\text{bull})$-free graphs, and $\bullet$ $(P_4+\ell P_1,P_5,\text{cricket})$-free graphs. In fact, all but the first of these are special cases of our general result that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,B_{4}(m),B_{3}(m)<sup>{+})$-free graphs for all $k\ge 1$ and $\ell,m\ge 0$. Here $B_{n}(m)$ is the graph obtained from a path of order $n$ by identifying one of its leaves with the centre vertex of $K_{1,m}$ and $B_{n}(m)<sup>{+}$ is the graph obtained by identifying an edge of $K_3$ with the edge of $B_{n}(m)$ with endpoints of degrees $2$ and $m$, respectively. Our results imply the existence of simple polynomial-time certifying algorithms to decide the $k$-colourability of all graphs in these subfamilies for every fixed $k$. We also show that $χ(G)\le \ell+2$ for all $(P_4+\ell P_1,K_3)$-free graphs and all $\ell\ge 0$, improving the previously known upper bound of $2\ell+2$ that followed from Randerath and Schiermeyer's 2004 result on $(P_t,K_3)$-free graphs. More generally, we provide a $χ$-bound in $O(\ell<sup>{ω-1})$ for $(P_4+\ell P_1)$-free graphs which improves the bound of $(2\ell+2)<sup>{ω-1}$ which followed from Gravier, Hoàng and Maffray in 2003 for $P_{t}$-free graphs.

• The paper proves that for any k and ℓ, there exist only finitely many k-vertex-critical graphs within several subclasses of (P₄+ℓP₁)-free graphs.
• It employs structural analysis and combinatorial techniques, including a new lemma based on Sperner's Theorem, to establish improved chromatic bounds.
• The results yield practical, polynomial-time certifying algorithms for k-colorability, enhancing algorithmic reliability in these graph families.

Vertex-Critical Graphs in $(P_4+\ell P_1)$-Free Graphs: New Finiteness Results

Introduction and Background

The classification of vertex-critical graphs—graphs that require $k$ colors for proper vertex-coloring but drop in chromatic number when any vertex is removed—is a core problem in structural and algorithmic graph theory, particularly within hereditary classes defined by forbidden induced subgraphs. The central focus of this paper is the family of $(P_4+\ell P_1)$-free graphs, i.e., graphs without an induced subgraph comprised of a $P_4$ (a path of four vertices) alongside $\ell$ isolated vertices.

Prior work established strong polynomial-time colorability and finiteness results for $k$-vertex-critical graphs under restrictions disallowing certain paths or linear forests, for example in $P_t$-free or $(P_5+\ell P_1)$-free graphs. However, finiteness of $k$-vertex-critical graphs in the case of $(P_4+\ell P_1)$-free graphs for arbitrary $k$0 and $k$1 had remained open except in limited cases with small values of $k$2 or with additional forbidden induced subgraphs (such as co-gem or paw+$k$3). This paper addresses the open question by targeting specific, structurally defined subfamilies of $k$4-free graphs, providing new positive finiteness results.

Main Results

The central contributions are finiteness results for $k$5-vertex-critical graphs in several nontrivial intersecting hereditary subfamilies:

1. Finiteness in $k$6-free graphs: The paper shows that for any $k$7 and $k$8, there are only finitely many $k$9-vertex-critical graphs within the class of graphs excluding both $(P_4+\ell P_1)$0 and two disjoint edges ($(P_4+\ell P_1)$1) as induced subgraphs.
2. Generalized finiteness for $(P_4+\ell P_1)$2-free graphs: The authors prove that for any $(P_4+\ell P_1)$3 and all $(P_4+\ell P_1)$4, only finitely many $(P_4+\ell P_1)$5-vertex-critical graphs exist when, in addition to $(P_4+\ell P_1)$6, certain structured graphs $(P_4+\ell P_1)$7 and $(P_4+\ell P_1)$8 are forbidden. Here, $(P_4+\ell P_1)$9 denotes a graph formed by attaching a $P_4$0 to a leaf of $P_4$1, and $P_4$2 is constructed by coalescing $P_4$3 and $P_4$4 along a specified edge.
3. Consequential subclass results: The general theorem yields corollaries establishing finiteness of $P_4$5-vertex-critical graphs in additional hereditary classes, such as $P_4$6-free, $P_4$7-free, and $P_4$8-free graphs for all $P_4$9 and $\ell$0.

The proofs leverage a new technical lemma concerning the structure of $\ell$1-free graphs, coupling Sperner's Theorem with the theory of antichains and independence in $\ell$2-vertex-critical contexts. The results exploit the existing finiteness in $\ell$3-free graphs and carefully analyze the impact of large independent sets and colorability constraints.

Chromatic Bounds

The paper strengthens earlier general chromatic bounds for $\ell$4-free graphs. They establish that for every $\ell$5-free graph, the chromatic number satisfies the sharp upper bound $\ell$6, improving the previously best-known bound of $\ell$7 for this class. More generally, for $\ell$8-free graphs the chromatic number admits an improved $\ell$9 upper bound, where $k$0 is the clique number, refining the $k$1 result of Gravier, Hoàng, and Maffray [GRAVIER2003].

These chromatic bounds not only delineate the structural limitations imposed by the forbidden induced subgraphs but also directly imply algorithmic consequences for vertex coloring.

Certifying Algorithms and Algorithmic Implications

One of the important algorithmic implications of these finiteness results is the existence of simple, polynomial-time certifying algorithms for $k$2-colorability in these subfamilies. That is, for any fixed $k$3, one can check $k$4-colorability by searching for the finite set of $k$5-vertex-critical forbidden induced subgraphs, and if one is found, produce it as a certificate of non-$k$6-colorability. This leads to robust, certifying algorithms—algorithms that not only provide a solution but also a verifiable witness—enhancing algorithmic reliability and trust [McConnell2011].

Context within the Literature

The results represent a significant advance in the incremental project of classifying when hereditary graph classes (defined by small, disconnected forbidden induced subgraphs) admit only finitely many $k$7-vertex-critical graphs for all $k$8. The finiteness and colorability of $k$9-free and $P_t$0-free graphs are now well-understood for $P_t$1 in many cases, but this work targets structural subclasses of the highly intractable $P_t$2-free case, particularly for larger $P_t$3.

The structural lemmas and techniques introduced, particularly those regarding the non-neighborhoods of large independent sets in $P_t$4-free graphs and their multipartite extensions, are likely to be adaptable to further open cases and may accelerate resolution of currently open dichotomy questions, such as the finiteness of $P_t$5-vertex-critical $P_t$6-free graphs for all $P_t$7 and $P_t$8 [CameronHoangSawada2022, KCameron2021].

Open Problems and Future Directions

The paper concludes with several pertinent open problems:

• Extension to $P_t$9-free graphs: Whether the restriction to $(P_5+\ell P_1)$0 and $(P_5+\ell P_1)$1 can be lifted, fully resolving the finiteness in $(P_5+\ell P_1)$2-free graphs, remains open.
• Finiteness in $(P_5+\ell P_1)$3-free graphs: Fully characterizing the finiteness for all $(P_5+\ell P_1)$4 would complete the dichotomy for triangle-free settings.
• Sharpness of chromatic bounds: Determining the optimal $(P_5+\ell P_1)$5 such that all $(P_5+\ell P_1)$6-free graphs satisfy $(P_5+\ell P_1)$7.
• Algorithm design in related classes: Developing certifying algorithms in further open classes, potentially by reducing to the finite subcases shown in this work.

These directions are closely tied to structural characterization, colorability bounds, and algorithmic tractability within hereditary classes, and will inform both complexity-theoretic and structural advances in graph coloring theory.

Conclusion

This paper achieves substantial progress in the classification of $(P_5+\ell P_1)$8-vertex-critical graphs within subfamilies of $(P_5+\ell P_1)$9-free graphs, providing broad and new finiteness results for several nontrivial classes by means of sophisticated structural analysis and combinatorial reasoning. The improvements in chromatic number bounds and the establishment of certifying algorithmic frameworks further broaden the practical applicability of these results. The open problems outlined form a clear agenda for the continued exploration of critical graph structures, their coloring properties, and associated algorithmic consequences in hereditary graph classes.

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References

• McConnell, R. M., et al., "Certifying algorithms" [McConnell2011].
• Gravier, S., Hoàng, Ch.T., Maffray, F., "Coloring the hypergraph of maximal cliques of a graph with no long path" [GRAVIER2003].
• Cameron, B., Hoàng, Ch.T., Sawada, J., "Dichotomizing $k$0-vertex-critical $k$1-free graphs for $k$2 of order four" [CameronHoangSawada2022].
• Cameron, K., Goedgebeur, J., Huang, S., Shi, Y., "$k$3-Critical graphs in $k$4-free graphs" [KCameron2021].