A Probabilistic Approach to Problems on Distance Graphs and Graphs of Diameters (Candidate-Degree Dissertation Author's Review, in Russian)

TL;DR

This dissertation explores probabilistic properties of distance graphs and diameter graphs, establishing thresholds for their structure and colorability in random graph models, with implications for combinatorial geometry.

Contribution

It provides new probabilistic thresholds for unit-distance graphs and diameter graphs in random models, advancing understanding of their chromatic properties in various dimensions.

Findings

Induced subgraphs with chromatic number ≤ 4 cover at least 91.7% of vertices in plane unit-distance graphs.

Threshold probability for a random graph to be isomorphic to a unit-distance graph is Θ(1/n) for dimensions 2-8.

Graphs of diameters tend to be bipartite when p is small and complete when p is large.

Abstract

The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced subgraph with chromatic number at most 4 that covers at least 91.7 percent of the vertices of the whole graph. The results of the second and third parts are related to the standard model of a random graph with n labeled vertices in which the edges occur independently with probability p, where p is a function of n. This is known as the Erdos--Renyi model G(n,p). Given a monotone property of a graph, Erdos and Renyi's theorem (1960) states that there exists a critical threshold value of p(n) below which the probability that a random graph has that property tends to one (as n tends to infinity) and above which the probability tends to zero. The main results of the second part are concerned with the (monotone) property of a graph to be isomorphic to some unit-distance graph in Euclidean d-space with fixed dimension d. The results of this part guarantee that for d in {2, 3, 4, 5, 6, 7, 8}, the threshold value of p(n) is big-Theta of 1/n. Furthermore, the case d = 1 stands apart from the case of higher dimensions; here the threshold probability is big-Theta of 1/(n^(4/3)). The results of the third part are devoted to studying "graphs of diameters" from the probabilistic standpoint. In particular, it is shown that under some conditions, almost all graphs of diameters in the plane have chromatic number less than 3. More generally, it is shown for G(n,p) that graphs of diameters have a tendency to chromatic degeneration (for large n) when p is close to 0, but have a tendency to completeness when p is close to 1.