Scott's Induced Subdivision Conjecture for Maximal Triangle-Free Graphs

Scott conjectured in [6] that the class of graphs with no induced subdivision of a given graph is χ-bounded. We verify his conjecture for maximal triangle-free graphs. Let F be a graph. We denote by Forb∗(F ) the class of graphs with no induced subdivision of F . A class G of graphs is χ-bounded if there exists a function f such that every graph G of G satisfies χ(G) ≤ f(ω(G)), where χ and ω respectively denote the chromatic number and the clique number of G. Gyárfás conjectured that Forb∗(F ) is χ-bounded if F is a cycle [3]. Scott proved that for each tree T , Forb∗(T ) is χ-bounded and conjectured the following [6]. Conjecture 1. For every graph F , Forb∗(F ) is χ-bounded. This question is open for triangle-free graphs, which is probably the core of the problem. It also has nice corollaries, for instance it would imply that any collection of segments in the plane with no three of them pairwise intersecting can be partitioned into a bounded number of non intersecting sets of segment. This is a well-known question of Erdős, first cited in [3]. Our goal is to prove Scott’s conjecture for triangle-free graphs with diameter two, i.e. maximal triangle-free graphs. Theorem 2. Let F be a graph of size l. Every maximal triangle-free graph G with χ(G) ≥ eθ(l4) contains an induced subdivision of F . Proof. Let H be the neighborhood hypergraph of G, i.e. the hypergraph with vertex set V and with hyperedges the closed neighborhoods of the vertices of G. Observe that H has packing number one, i.e. its hyperedges pairwise intersect. Note also that if the transversality of H is t (minimum size of a set of vertices intersecting all hyperedges), then χ(G) ≤ 2t. Indeed, G can be covered by t closed neighborhoods, hence by t induced stars since G is triangle-free. Since χ(G) ≥ eθ(l4), the transversality of H is at least eθ(l 4). Ding, Seymour and Winkler [2] proved that if a hypergraph H has packing number one and transversality greater than 11d2(d + 4)(d + 1)2, it contains d hyperedges e1, . . . , ed and a set of vertices Y = {yi,j : 1 ≤ i < j ≤ d} such that yi,j ∈ ei ∩ ej and yi,j / ∈ ek for all k 6= i, j. Since the transversality of H is at least eθ(l 4), we have such a collection of hyperedges e1, . . . , ed with d ≥ eθ(l4).