# Maximum Size of a Family of Pairwise Graph-Different Permutations

## TL;DR

This paper investigates the maximum size of families of permutations that are pairwise different with respect to a graph's structure, providing bounds and exact values for various classes of graphs, especially bipartite graphs.

## Contribution

It establishes bounds on the maximum size of pairwise graph-different permutation families for balanced bipartite graphs and explores extensions involving disjoint vertices.

## Key findings

- Maximum size for balanced bipartite graphs is approximately 2^{n}
- Examples of bipartite graphs with low maximum degree achieving similar bounds
- Exact value determined for the graph of two disjoint edges

## Abstract

Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise $G$-different permutations for various graphs $G$. We show that for all balanced bipartite graphs $G$ of order $n$ with minimum degree $n/2 - o(n)$, the maximum number of pairwise $G$-different permutations of the vertices of $G$ is $2^{(1-o(1))n}$. We also present examples of bipartite graphs $G$ with maximum degree $O(\log n)$ that have this property. We explore the problem of bounding the maximum size of a family of pairwise graph-different permutations when an unlimited number of disjoint vertices is added to a given graph. We determine this exact value for the graph of 2 disjoint edges, and present some asymptotic bounds relating to this value for graphs consisting of the union of $n/2$ disjoint edges.