Twins in graphs

TL;DR

This paper investigates the existence and size of twin vertex subsets in graphs, extending the pigeonhole principle to complex structures and providing bounds based on graph properties.

Contribution

It introduces the concept of twins in graphs, establishes bounds on their size using discrepancy results, and characterizes conditions for maximum twin size.

Findings

Bounds on twin size in terms of edges and vertices

Conditions for twins of size half the vertices

Twin size in forests is at least half minus one

Abstract

A basic pigeonhole principle insures an existence of two objects of the same type if the number of objects is larger than the number of types. Can such a principle be extended to a more complex combinatorial structure? Here, we address such a question for graphs. We call two disjoint subsets $A, B$ of vertices \emph{\textbf{twins}} if they have the same cardinality and induce subgraphs of the same size. Let $t(G)$ be the largest $k$ such that $G$ has twins on $k$ vertices each. We provide the bounds on $t(G)$ in terms of the number of edges and vertices using discrepancy results for induced subgraphs. In addition, we give conditions under which $t(G)= |V(G)|/2$ and show that if $G$ is a forest then $t(G) \geq |V(G)|/2 - 1$.