Achromatic number, achromatic index and diachromatic number of circulant graphs and digraphs

TL;DR

This paper determines the achromatic and diachromatic numbers of certain circulant graphs and digraphs, providing exact values and bounds, and explores their relation to projective cyclic planes.

Contribution

It introduces new exact formulas and bounds for the achromatic and diachromatic numbers of circulant graphs and digraphs with two lengths, including special cases related to projective cyclic planes.

Findings

Exact achromatic number for specific circulant graphs: 8q+5.

Exact diachromatic number for specific circulant digraphs: 8q+3.

Lower bounds for general circulant graphs and digraphs with two lengths.

Abstract

In this paper, we determine the achromatic and diachromatic numbers of some circulant graphs and digraphs each one with two lengths and give bounds for other circulant graphs and digraphs with two lengths. In particular, for the achromatic number we state that $\alpha(C_{16q^2+20q+7}(1,2))=8q+5$, and for the diachromatic number we state that $dac(\overrightarrow{C}_{32q^2+24q+5}(1,2))=8q+3$. In general, we give the lower bounds $\alpha(C_{4q^2+aq+1}(1,a))\geq 4q+1$ and $dac(\overrightarrow{C}_{8q^2+2(a+4)q+a+3}(1,a))\geq 4q+3$ when $a$ is a non quadratic residue of $\mathbb{Z}_{4q+1}$ for graphs and $\mathbb{Z}_{4q+3}$ for digraphs, and the equality is attained, in both cases, for $a=3$. Finally, we determine the achromatic index for circulant graphs of $q^2+q+1$ vertices when the projective cyclic plane of odd order $q$ exists.