Flips in Graphs

TL;DR

This paper investigates a graph parameter related to quantum-error-correcting codes, providing asymptotic estimates for random graphs and bounds on the maximum value of this parameter for graphs of size n.

Contribution

It introduces and analyzes a new graph parameter, offering tight asymptotic estimates for random graphs and bounds on its maximum over all graphs of a given size.

Findings

Asymptotically tight estimates of f(G) for G_{n,p} when p is constant.

Upper bound of (0.382+o(1))n for the maximum of f(G) over graphs with n vertices.

Connections to quantum-error-correcting codes and combinatorial graph properties.

Abstract

We study a problem motivated by a question related to quantum-error-correcting codes. Combinatorially, it involves the following graph parameter: $$f(G)=\min\set{|A|+|\{x\in V\setminus A : d_A(x)\text{is odd}\}| : A\neq\emptyset},$$ where $V$ is the vertex set of $G$ and $d_A(x)$ is the number of neighbors of $x$ in $A$. We give asymptotically tight estimates of $f$ for the random graph $G_{n,p}$ when $p$ is constant. Also, if $$f(n)=\max\set{f(G): |V(G)|=n}$$ then we show that $f(n)\leq (0.382+o(1))n$.