The Generalised Shift Graph

TL;DR

This paper extends the concept of Shift Graphs to include various arrangements called types, focusing on specific types and their properties, such as isomorphism, chromatic number, and automorphisms, especially over ordinals.

Contribution

It introduces a generalized framework for Shift Graphs based on types, analyzes their isomorphism conditions over ordinals, and explores their chromatic and automorphism properties.

Findings

Isomorphism of graphs implies equality of ordinals for certain types.

Studied chromatic number of the generalized graphs.

Analyzed automorphism groups of these graphs.

Abstract

In 1968, Erd\"os defined the Shift Graph as the graph whose vertices are the $k$-element subsets of $[n]=\{0,1,2,...,n-1\}$ such that $A=\{a_1,...,a_k\}$ and $B=\{b_1,...,b_k\}$ are neighbours iff $a_1<b_1=a_2<b_2=a_3<... <b_{n-1}=a_n<b_n$. In the paper \textit{On the Generalised Shift Graph}, Avart, Luczac and R\"odl extend this definition to include all possible arrangements of the $a_is$ and $b_is$, known as \textit{types}. In this paper, we will consider a selection of these types and study the corresponding graphs. We are interested in to what extent the graphs $G(S,\tau)$ and $G(S',\tau)$ are distinct for distinct linear orderings $S,S'$ and for some type $\tau$. In this paper, we will concentrate on ordinals and types of the form $\sigma_{a,b}=11...133...322...2$. We will show that if $G(\alpha,\sigma_{a,b})\cong G(\beta,\sigma_{a,b})$ then $\alpha=\beta$. We will also consider the chromatic number and the automorphism groups of these graphs in order to gain a deeper understanding of their properties.

Full text

More on the Generalised Shift Graph

Milette Riis

The Generalised Shift Graph

M. Riis (University of Leeds, UK)

Abstract

In 1968, Erdös111[shiftnew] attributes this to [erdos] defined the Shift Graph as the graph whose vertices are the $k$-element subsets of $[n]=\{0,1,2,...,n-1\}$ such that $A=\{a_{1},...,a_{k}\}$ and $B=\{b_{1},...,b_{k}\}$ are neighbours iff $a_{1}<b_{1}=a_{2}<b_{2}=a_{3}<...<b_{n-1}=a_{n}<b_{n}$. In the paper On the Generalised Shift Graph, Avart, Luczac and Rödl extend this definition to include all possible arrangements of the $a_{i}s$ and $b_{i}s$, known as types. In this paper, we will consider a selection of these types and study the corresponding graphs.

We are interested in to what extent the graphs $G(S,\tau)$ and $G(S^{\prime},\tau)$ are distinct for distinct linear orderings $S,S^{\prime}$ and for some type $\tau$. In this paper, we will concentrate on ordinals and types of the form $\sigma_{a,b}=11...133...322...2$. We will show that if $G(\alpha,\sigma_{a,b})\cong G(\beta,\sigma_{a,b})$ then $\alpha=\beta$. We will also consider the chromatic number and the automorphism groups of these graphs in order to gain a deeper understanding of their properties.

1 Background Material

We would like to describe a new class of graphs. Each graph is determined by two things: a totally ordered set $S$, and a “type” $\tau$; we thus denote these graphs by $G(S,\tau)$. Both the edges and the vertices of $G$ are encoded by $\tau$.