The Generalised Shift Graph

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.