Full Proof of the Existence of a Degree 8 Circulant graph of Order L(8,k) of Arbitrary Diameter k

TL;DR

This paper provides a comprehensive proof confirming the existence of the largest known degree 8 circulant graphs for all diameters, extending previous work and following established proof techniques.

Contribution

It offers the full proof of the existence of degree 8 circulant graphs for all diameters, including detailed exception cases, building on prior methods used for degree 6 graphs.

Findings

Confirmed existence of degree 8 circulant graphs for all diameters

Extended proof techniques from degree 6 to degree 8

Included full exception cases in the proof

Abstract

This is the full proof of Theorem 3 on the existence of the largest known degree 8 circulant graph for all diameters stated in the paper "The degree-diameter problem for circulant graphs of degree 8 and 9" by the author. To avoid the paper being unduly long the exceptions for only one case were included in the statement of the proof. In the statement of the proof presented in this paper the exceptions for all cases are included in full. This proof closely follows the approach taken by Dougherty and Faber for the existence of the largest known degree 6 circulant graph for all diameters.