A positive instance of Scott's Conjecture on induced subdivisions

For a graph $G$, $χ(G)$ denotes the chromatic number of $G$ and $ω(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $χ$-bounded if there is a function $f$ such that for each graph $G$ in the class, $χ(G) \le f(ω(G))$. Scott (1997) conjectured that for every graph $H$, the class of graphs which do not contain any subdivision of $H$ as an induced subgraph is $χ$-bounded. He proved his conjecture when $H$ is a tree and when $H$ is the complete graph on four vertices, $K_4$. Esperet and Trotignon (2019) proved that the conjecture holds when $H$ is $K_4$ with one edge subdivided once. Scott’s conjecture was disproved by Pawlik et al. (2014). Chalopin et al. (2016) gave more counterexamples including the graph obtained from $K_4$ by subdividing each edge of a 4-cycle once. We prove that the conjecture holds when $H$ consists of a complete bipartite graph with and additional vertex which has exactly two neighbours, on the same side of the bipartition. As a special case, this proves Scott’s conjecture when $H$ is obtained from $K_4$ by subdividing two disjoint edges.

💡 Research Summary

The paper addresses Scott’s 1997 conjecture, which posits that for every graph H, the class of graphs that contain no induced subdivision of H is χ‑bounded (i.e., the chromatic number can be bounded by a function of the clique number). While the conjecture is known to hold for trees, K₄, and K₄ with a single subdivided edge, it fails for many graphs containing cycles, as shown by Pawlik et al. (2014) and Chalopin et al. (2016). The authors identify a new family of graphs H for which the conjecture does hold, thereby providing a positive instance beyond the previously known cases.

The target graphs are denoted P₍ₐ,ᵦ₎. They consist of a complete bipartite graph K₍ₐ₊₂,ᵦ₊₂₎ together with an extra vertex v that is adjacent to exactly two vertices x and y lying on the same side of the bipartition (the side of size β + 2). When a = b, the graph is symmetric; the special case P₂,₁ is isomorphic to K₄ with two disjoint edges each subdivided once (often written K₄^{++}). The main theorem proves that for every integer a ≥ 2, the class Cₐ of graphs that contain no induced subdivision of P₍ₐ,ₐ₎ is χ‑bounded. Consequently, the conjecture holds for K₄^{++} as a corollary.

The proof combines several deep tools:

1. Average‑degree forcing – Theorem 1 (Kühn‑Osthus) and its stronger polynomial‑bound version (Theorem 2, Bourneuf et al., Girão‑Hunter) guarantee that any graph of sufficiently large average degree either contains an induced subdivision of P₍ₐ,ₐ₎ or a large complete bipartite subgraph K_{s,s}. This allows the authors to focus on graphs that contain a large bipartite “core”.

2. Template construction – Starting from a large K_{s,s}, the authors iteratively grow a complete r‑partite subgraph X = (X₁,…,Xᵣ) (the “template”), where each part has size at least a + 1. The template is maximal in the sense that no further part can be added without creating an induced P₍ₐ,ₐ₎.

3. Attachment analysis – Lemma 1 and Lemma 2 describe how a vertex outside the template can be adjacent to the parts of X. In particular, a vertex can have at most one part where it has many non‑neighbours; otherwise it would create a forbidden P₍ₐ,ₐ₎. Lemma 3 shows that if there are many vertices that are “(a‑1)‑disconnected” from every part, the template can be extended, contradicting maximality. Hence the set Z of such vertices is bounded.

4. Partition of the remaining vertices – The vertices outside X ∪ Z are split into three groups: * A: vertices 1‑connected to every part, * C: vertices that have a unique “conflict” part containing at least a + 1 non‑neighbours, * M: vertices that are 1‑connected to at least two parts but (a‑1)‑disconnected from at least one part.

   Two main cases are considered: If A is empty, the set S consisting of a + 1 vertices from each part of X forms a dominating set. Each vertex of S has neighbourhood clique number at most ω(G) − 1, so by the induction hypothesis it can be coloured with τ colours. Adding a unique colour for each vertex of S yields a colouring using at most (1 + τ)(a + 1) ω(G) colours. If A is non‑empty, each connected component of A can be separated from the template by deleting a small cutset whose size is bounded by a function of ω(G) and the parameters a, f (where f = a(ω(G)+1)+2). Nguyen’s connectivity theorem (Theorem 3) then guarantees that any subgraph with chromatic number exceeding this bound must contain a highly connected subgraph, which would force an induced P₍ₐ,ₐ₎ – a contradiction. Hence the whole graph can be coloured with a number of colours bounded by an explicit expression involving d(P₍ₐ,ₐ₎, s), R(·,·) (a Ramsey‑type function derived from the Erdős‑Hajnal result for P₍ₐ,ₐ₎‑free graphs), and ω(G).

The final bound (Theorem 6) is rather intricate: it combines the degree threshold d(P₍ₐ,ₐ₎, R(ω, f)), the Ramsey‑type term R(ω, f), and linear terms in ω(G). Nevertheless, it is a concrete function fₐ(·) such that χ(G) ≤ fₐ(ω(G)) for every G ∈ Cₐ.

By setting a = 2, the authors obtain that the class of graphs without an induced subdivision of K₄^{++} (K₄ with two disjoint edges each subdivided once) is χ‑bounded, thereby confirming Scott’s conjecture for this previously unresolved graph. More generally, any graph formed by a complete bipartite graph together with a vertex adjacent to exactly two vertices on the same side of the bipartition also satisfies the conjecture.

In summary, the paper extends the list of graphs for which Scott’s conjecture holds, introduces a robust template‑based framework for handling induced subdivisions, and leverages recent advances in average‑degree forcing, high‑connectivity, and Erdős‑Hajnal phenomena. The techniques are likely to be applicable to further families of graphs, offering a promising direction for future research on χ‑boundedness of induced‑subdivision‑free classes.