# 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.

## Key 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.