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Interval Graph Limits.

Persi Diaconis1, Susan Holmes2, Svante Janson3

  • 1Department of Mathematics, Stanford University, Stanford, CA 94305, USA.

Annals of Combinatorics
|September 26, 2015
PubMed
Summary
This summary is machine-generated.

This study introduces a new graph limit theory for dense interval graphs by altering coordinate distributions, not the typical symmetric function. This approach yields continuous graph limits and offers insights into random interval graphs and graph properties.

Keywords:
graph limitsintersection graphsinterval graphs

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Area of Science:

  • Graph theory
  • Combinatorics
  • Probability theory

Background:

  • Graph limit theory typically uses symmetric functions on a unit square.
  • Dense interval graphs are a significant class of graphs with unique properties.

Purpose of the Study:

  • To develop a novel graph limit theory tailored for dense interval graphs.
  • To explore alternative approaches to graph limits by modifying probability distributions.
  • To investigate the continuity of graph properties within this new framework.

Main Methods:

  • Developing a graph limit theory that fixes the function W(x, y) and varies the underlying probability distributions of coordinates x and y.
  • Analyzing the continuity of graph limits under these modified distributions.
  • Establishing connections to random interval graphs with illustrative examples.

Main Results:

  • A new graph limit theory for dense interval graphs is established.
  • Continuous graph limits are achieved through specific choices of coordinate distributions.
  • Continuity results for chromatic and clique numbers of interval graphs are demonstrated.
  • Uniqueness results for general graph limit descriptions are presented.

Conclusions:

  • The developed theory offers a novel perspective on graph limits, particularly for dense interval graphs.
  • The findings provide a deeper understanding of random interval graphs and their properties.
  • The continuity results have implications for analyzing graph parameters in limit contexts.