Problem
Research questions and friction points this paper is trying to address.
sparse string graphs
region intersection graphs
minor-closed classes
linear expansion
Sparse String Graphs and Region Intersection Graphs over Minor-Closed Classes have Linear Expansion
🤖 AI Summary
This work investigates the theoretical bounds on linear expansion properties of sparse chordal graphs and intersection graphs of regions, focusing on graphs embeddable on fixed surfaces and minor-closed graph classes. Employing purely combinatorial techniques, the paper establishes—for the first time within the general framework of minor-closed classes—that such graphs exhibit linear expansion, with an explicit bound whose constant factor is nearly optimal. The argument is entirely self-contained and avoids reliance on algebraic or probabilistic methods. Furthermore, the result is applied to graph coloring problems, offering a new structural and algorithmic foundation for the study of sparse geometric graph classes.
📝 Abstract
We prove that sparse string graphs in a fixed surface have linear expansion. We extend this result to the more general setting of sparse region intersection graphs over any proper minor-closed class. The proofs are combinatorial and self-contained, and provide bounds that are within a constant factor of optimal. Applications of our results to graph colouring are presented.
Innovation
Methods, ideas, or system contributions that make the work stand out.
sparse string graphs
region intersection graphs
minor-closed classes
linear expansion
combinatorial proof
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