[Paper Review] Restricted frame graphs and a conjecture of Scott

Jérémie Chalopin, Louis Esperet+2 more|arXiv (Cornell University)|Jun 2, 2014

Advanced Graph Theory Research8 references2 citations

TL;DR

This paper investigates Scott's conjecture on χ-boundedness of graphs excluding subdivisions of a fixed graph H as induced subgraphs. By analyzing the construction of triangle-free segment intersection graphs with unbounded chromatic number, the authors identify new counterexamples: any ≥2-subdivision of a 2-connected multigraph with no vertex in every cycle is a counterexample. They further show that Scott’s conjecture fails for all graphs obtained by subdividing every edge of K₄ at least once.

ABSTRACT

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once. It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.

Motivation & Objective

• To determine which graphs H violate Scott’s conjecture by failing to produce χ-bounded classes when excluding all subdivisions of H as induced subgraphs.
• To extend the known counterexamples to Scott’s conjecture beyond ≥1-subdivisions of non-planar graphs.
• To characterize the class of graphs H for which no subdivision of H appears as an induced subgraph in the Pawlik et al. construction of triangle-free segment intersection graphs.
• To prove that ≥2-subdivisions of 2-connected multigraphs with no vertex in every cycle are counterexamples to Scott’s conjecture.
• To establish a connection between restricted frame graphs and the structural properties of H that lead to counterexamples.

Proposed method

• Analyzing the Pawlik et al. construction of triangle-free intersection graphs of line segments in the plane with unbounded chromatic number.
• Using the fact that these graphs are also restricted frame graphs—defined as intersection graphs of specific arcwise connected shapes in the plane.
• Characterizing graphs H such that no subdivision of H can be represented as a restricted frame graph, implying such H are counterexamples to Scott’s conjecture.
• Applying a recursive algorithm to test whether a multigraph G admits a ≥2-subdivision that is a restricted frame graph, based on decomposition into 2-connected components.
• Using structural graph theory: identifying chandeliers, feedback vertices, and cut-vertices to determine when a ≥2-subdivision cannot be a restricted frame graph.
• Leveraging the stability of restricted frame graphs under twin addition, as shown in Walczak’s modified construction.

Experimental results

Research questions

• RQ1 For which graphs H is the class of graphs excluding all subdivisions of H as induced subgraphs not χ-bounded?
• RQ2 Can the construction of Pawlik et al. be used to generate a larger class of counterexamples to Scott’s conjecture than just ≥1-subdivisions of non-planar graphs?
• RQ3 Which multigraphs G have the property that every ≥2-subdivision of G is a counterexample to Scott’s conjecture?
• RQ4 What structural conditions on a multigraph G ensure that no ≥2-subdivision of G is a restricted frame graph?
• RQ5 Is there a complete characterization of graphs H such that no subdivision of H appears as an induced subgraph in the segment intersection construction?

Key findings

• Scott’s conjecture is false for any graph obtained by subdividing every edge of K₄ at least once.
• Any ≥2-subdivision of a 2-connected multigraph with no vertex contained in every cycle is a counterexample to Scott’s conjecture.
• The class of restricted frame graphs is closed under the addition of twins, which allows extending counterexamples from the original Pawlik et al. construction to Walczak’s modified construction.
• A complete algorithmic characterization is provided for when a multigraph G has a ≥2-subdivision that is a restricted frame graph, based on its 2-connected components and feedback vertices.
• The construction of Pawlik et al. yields graphs with unbounded chromatic number and no stable sets of linear size, implying that even the fractional chromatic number is unbounded in such families.
• The results show that the class of graphs excluding all subdivisions of a given H as induced subgraphs is not χ-bounded for a significantly larger family of H than previously known, including all ≥2-subdivisions of 2-connected multigraphs without a universal vertex.