On The Circular Altitude of Graphs

TL;DR

This paper explores properties of the circular altitude of graphs, establishing its behavior under graph blocks, homomorphisms, and Cartesian products, and clarifies its relationship with other graph parameters.

Contribution

It proves that the circular altitude equals the maximum of its blocks' altitudes, is invariant under homomorphisms, and equals the maximum of the factors' altitudes in Cartesian products.

Findings

Circular altitude equals maximum of block altitudes.

Homomorphically equivalent graphs share the same circular altitude.

Circular altitude of Cartesian product equals maximum of the factors' altitudes.

Abstract

Peter Cameron introduced the concept of the circular altitude of graphs; a parameter which was shown by Bamberg et al. that provides a lower bound on the circular chromatic number. In this note, we investigate this parameter and show that the circular altitude of a graph is equal to the maximum of circular altitudes of its blocks. Also, we show that homomorphically equivalent graphs have the same circular altitudes. Finally, we prove that the circular altitude of the Cartesian product of two graphs is equal to the maximum of circular altitudes of its factors.