Guessing Numbers and Extremal Graph Theory

TL;DR

This paper explores the guessing number of graphs, relating it to extremal graph theory and forbidden subgraph properties, providing bounds and construction methods for graphs with constrained guessing numbers.

Contribution

It establishes the extremal number and saturation number for graphs with bounded guessing number and links the property to forbidding finite subgraph sets.

Findings

Derived extremal number for graphs with guessing number ≤ a

Established upper bounds on saturation numbers

Provided methods to construct saturated graphs between bounds

Abstract

For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the colors of the vertices in its neighborhood. This quantity is related to problems in network coding, circuit complexity and graph entropy. We study the guessing number of graphs as a graph property in the context of classic extremal questions, and its relationship to the forbidden subgraph property. We find the extremal number with respect to the property of having guessing number $\leq a$, for fixed $a$. Furthermore, we find an upper bound on the saturation number for this property, and a method to construct further saturated graphs that lie between these two extremes. We show that, for a fixed number of colors, bounding the guessing number is equivalent to forbidding a finite set of subgraphs.