Recursive Relationships in the Classes of Odd Graphs and Middle Levels Graphs

TL;DR

This paper establishes recursive relationships and embeddings between odd graphs and middle levels graphs, enabling path lifting and revealing connections to Catalan numbers, thus advancing understanding of their structure and Hamiltonicity.

Contribution

It introduces a recursive framework linking odd graphs and middle levels graphs through embeddings, facilitating path lifting and connecting these graphs to Catalan numbers.

Findings

Middle levels graphs can be embedded in odd graphs and higher degree middle levels graphs.

The embedding allows recursive path lifting between graph classes.

Connections to Catalan numbers emerge from the graph relationships.

Abstract

The classes of odd graphs $O_n$ and middle levels graphs $B_n$ form one parameter subclasses of the Kneser graphs and bipartite Kneser graphs respectively. In particular both classes are vertex transitive while resisting definitive conclusions about their Hamiltonicity, and have thus come under scrutiny with regards to the Lov\'asz conjecture. In this paper, we will establish that middle levels graphs may always be embedded in odd graphs and middle levels graphs of higher degree, and furthermore, that this embedding allows us to define a recursion relationship in both classes which can be used to lift paths in $O_{n-1}$ (respectively $B_{n-1}$) to paths in $O_n$ (respectively $B_n$). This embedding also gives rise to the natural formation of a class of biregular graphs which give connections between the odd graphs, middle levels graphs and Catalan numbers.