Word-Representability of Shift Graphs

TL;DR

This paper proves that shift graphs, a class of sparse graphs with high chromatic number, are word-representable, and introduces generalized shift graphs with similar properties, highlighting differences between line graphs and line digraphs.

Contribution

The paper establishes that all shift graphs and their natural generalizations are word-representable, expanding understanding of word-representability in complex graph classes.

Findings

Shift graphs are word-representable.

Generalized shift graphs are also word-representable.

Line digraphs of certain word-representable graphs remain word-representable.

Abstract

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. For integers $n>k>0 $, the shift graph $G(n,k)$ is the graph whose vertex set consists of all increasing $k$-tuples $(x_1,x_2,\dots,x_k)$ with $1\le x_1<x_2<\cdots<x_k\le n$, where two vertices $(x_1,\dots,x_k)$ and $(y_1,\dots,y_k)$ are adjacent whenever $x_{i+1}=y_i$ for all $1\le i\le k-1$ or $y_{i+1}=x_i$ for all $1\le i\le k-1$. Shift graphs are classical examples of sparse graphs having arbitrarily high chromatic number and odd girth. We further observe that shift graphs arise naturally as induced subgraphs of simplified de Bruijn graphs. Although simplified de Bruijn graphs contain non-word-representable members in general, we prove that the entire class of shift graphs is word-representable. We also introduce a natural generalization of shift graphs in which adjacency is defined by more than one shift condition, and show that these generalized shift graphs are likewise word-representable. As a consequence, we obtain an explicit family of graphs exhibiting a contrast between line graph and line digraph constructions: there exists a family of word-representable graphs whose line graphs are not word-representable when the number of vertices is at least $5$, while their line digraphs are word-representable.