Structured Codes of Graphs

TL;DR

This paper determines the maximum size of graph families with edge set differences satisfying specific conditions, using extremal graph theory, and explores local conditions and related invariants.

Contribution

It provides complete solutions for certain graph family size problems under connectivity and Hamiltonicity constraints, introducing capacity invariants linked to chromatic numbers.

Findings

Exact maximum sizes for families with connectivity or Hamiltonicity conditions

Introduction of capacity invariants related to subgraph containment

Use of classical extremal graph theory results

Abstract

We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.