Some Properties of Alphabet Overlap Graphs

TL;DR

This paper studies the properties of alphabet overlap graphs, proving their Hamiltonian nature and determining their chromatic number under various conditions, thus advancing understanding of their structural characteristics.

Contribution

It establishes that alphabet overlap graphs are Hamiltonian for all non-trivial parameters and provides exact and bounded values for their chromatic number.

Findings

G is Hamiltonian for all non-trivial parameters

Exact chromatic number values when s ≥ k/2

Bounds on chromatic number when s < k/2

Abstract

Consider a graph G = G(k,d,s) with vertex set the set of all k-letter words over an alphabet of size d. An edge e = vw is in E iff v is distinct from w and the last(first) k-s letters of v are identical to the first(last) k-s letters of w. In this paper we show that G is Hamiltonian for all non-trivial values of the parameters and obtain exact values for its chromatic number when s is greater than or equal to k/2. We also obtain bounds when the chromatic number is less than k/2.