On types of growth for graph-different permutations

TL;DR

This paper investigates the maximum size of permutation sets with adjacency constraints defined by an infinite graph, providing exact results for specific cases and estimates for others, linking to Shannon capacity concepts.

Contribution

It characterizes the growth types of permutation sets constrained by graph adjacency and relates these to Shannon capacity within a specific context.

Findings

Exact cardinality for a special graph case

Estimates for other graph configurations

Comparison of results for complementary graphs

Abstract

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity "within a given type".