On the Chromatic Number of the \(\alpha\)-Overlap Graphs

Debra Knisley1, Yared Nigussie1, Attila Por2
1Department of Mathematics East Tennessee State University
2Department of Mathematics Western Kentucky University

Abstract

The generalized deBruijn graph \(dB(a,k)\) is the directed graph with \(a^k\) vertices and edges between vertices \(x = a_1, a_2, \ldots, a_k\) and \(y = b_1, b_2, \ldots, b_k\) precisely when \(a_2\cdots a_k = b_1,b_2\cdots b_{k-1}\). The deBruijn graphs can be further generalized by introducing an overlap variable \(t \leq k-1\) where the number of consecutive digits by which the vertex labels (sequences) overlap is \(t\). The \(\alpha\)-overlap graph is the underlying simple graph of the generalized deBruijn digraph with vertex label overlap \(0 t > 0\). Thus \(dB(a,k) = G(a,k,k-1)\). In this paper, we show that every \(\alpha\)-overlap graph is \(3\)-colorable for any \(a\) if \(k\) is sufficiently large. We also determine bounds on the chromatic number of the \(\alpha\)-overlap graphs if \(a\) is much larger than \(k\).