Linear embeddings of graphs and graph limits

• DOI: 10.1016/j.jctb.2015.02.002
• Publication Date: 2015-02-19T21:32:57Z
• Authors: Huda Chuangpishit; Mahya Ghandehari; Matthew Hurshman; Jeannette Janssen; Nauzer Kalyaniwalla
• Keywords: FOS: Computer and information sciences; Discrete Mathematics (cs.DM); Primary 46L07, 47B47; FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); 0102 computer and information sciences; 01 natural sciences; Computer Science - Discrete Mathematics

In press<br/>Consider a random graph process where vertices are chosen from the interval $[0,1]$, and edges are chosen independently at random, but so that, for a given vertex $x$, the probability that there is an edge to a vertex $y$ decreases as the distance between $x$ and $y$ increases. We call this a random graph with a linear embedding. We define a new graph parameter $��^*$, which aims to measure the similarity of the graph to an instance of a random graph with a linear embedding. For a graph $G$, $��^*(G)=0$ if and only if $G$ is a unit interval graph, and thus a deterministic example of a graph with a linear embedding. We show that the behaviour of $��^*$ is consistent with the notion of convergence as defined in the theory of dense graph limits. In this theory, graph sequences converge to a symmetric, measurable function on $[0,1]^2$. We define an operator $��$ which applies to graph limits, and which assumes the value zero precisely for graph limits that have a linear embedding. We show that, if a graph sequence $\{ G_n\}$ converges to a function $w$, then $\{ ��^*(G_n)\}$ converges as well. Moreover, there exists a function $w^*$ arbitrarily close to $w$ under the box distance, so that $\lim_{n\rightarrow \infty}��^*(G_n)$ is arbitrarily close to $��(w^*)$.<br/>