On the non-existence of extended perfect codes and some perfect colorings

TL;DR

This paper establishes necessary conditions for perfect colorings in certain graphs and uses these to prove the non-existence of extended perfect codes in specific Hamming and Doob graphs, advancing understanding of their combinatorial structure.

Contribution

It provides new necessary conditions for perfect colorings in Hamming and Doob graphs and proves the non-existence of extended perfect codes in these settings.

Findings

Necessary conditions for perfect colorings in Hamming and Doob graphs.

Non-existence of extended perfect codes in specified graphs.

Clarification of limitations for perfect code existence.

Abstract

In this paper we obtain the necessary condition for the existence of perfect $k$-colorings (equitable $k$-partitions) in Hamming graphs $H(n,q)$, where $q=2,3,4$ and Doob graphs $D(m,n)$. As an application, we prove the non-existence of extended perfect codes in $H(n,q)$, where $q=3,4$, $n>q+2$, and in $D(m,n)$, where $2m+n>6$.