Locally triangular graphs and normal quotients of the $n$-cube

TL;DR

This paper characterizes connected locally triangular graphs as halved graphs of normal quotients of n-cubes, extending known results about bipartite rectagraphs and exploring a generalized minimum distance parameter.

Contribution

It provides a new characterization of locally triangular graphs via normal quotients of n-cubes and introduces a generalized minimum distance concept for automorphism groups.

Findings

Connected locally triangular graphs are halved graphs of normal quotients of n-cubes.

A new parameter generalizing minimum distance is studied for automorphism groups of the n-cube.

Refinement of the relationship between rectagraphs and locally triangular graphs.

Abstract

For an integer $n\geq 2$, the triangular graph has vertex set the $2$-subsets of $\{1,\ldots,n\}$ and edge set the pairs of $2$-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every $2$-path lies in a unique quadrangle. We refine this result and provide a characterisation of connected locally triangular graphs as halved graphs of normal quotients of $n$-cubes. To do so, we study a parameter that generalises the concept of minimum distance for a binary linear code to arbitrary automorphism groups of the $n$-cube.