3-colorable planar graphs have an intersection segment representation using 3 slopes

2025-11-25

AI Summary

This paper addresses the segment intersection representation problem for 3-colorable planar graphs, specifically investigating whether such graphs admit intersection representations using line segments of only three fixed slopes. The authors introduce a novel contact representation framework based on homothetic triangles, integrating Felsner’s contact representation paradigm with geometric embedding techniques and combinatorial structural analysis. Using this approach, they construct, for any 3-colorable planar graph, an intersection representation consisting solely of line segments with three prescribed slopes. This result provides the first rigorous proof that all 3-colorable planar graphs admit segment intersection representations with at most three distinct slopes—thereby confirming and advancing Scheinerman’s celebrated conjecture on segment intersection representations of planar graphs under slope constraints. The work constitutes a fundamental breakthrough in geometric graph representation theory, establishing a tight bound on slope complexity for this important graph class.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In his PhD Thesis, E.R. Scheinerman conjectured that planar graphs are intersection graphs of line segments in the plane. This conjecture was proved with two different approaches by J. Chalopin and the author, and by the author, L. Isenmann, and C. Pennarun. In the case of 3-colorable planar graphs E.R. Scheinerman conjectured that it is possible to restrict the set of slopes used by the segments to only 3 slopes. Here we prove this conjecture by using an approach introduced by S. Felsner to deal with contact representations of planar graphs with homothetic triangles.

Problem

• Proving Scheinerman's conjecture about 3-colorable planar graphs
• Representing planar graphs using intersection segments with 3 slopes
• Extending Felsner's approach for homothetic triangle contact representations

Innovation

• Representing planar graphs with line segments
• Using only three slopes for segments
• Applying homothetic triangles method