Perfect codes and regular sets in vertex-transitive graphs
Alireza Abdollahi, Zeinab Akhlaghi, Majid Arezoomand
TL;DR
This work addresses regular sets and perfect codes in vertex-transitive graphs by expressing graphs as coset graphs $\mathrm{Cos}(G,H,U)$ and examining subgroup relations $H \le A \le G$. A key contribution is Theorem A, which gives a necessary and sufficient condition for $A$ to be an $(r,s)$-regular set of $(G,H)$ via an inverse-closed subset $X$ with $XH=HX^{-1}$ such that $XH \cap A$ is a union of $r$ left cosets of $H$ and $XH \cap tA$ is a union of $s$ left cosets for each $t \in G\setminus A$. Theorem B generalizes to $H \unlhd A \unlhd G$ and yields divisibility/intersection criteria; Theorem C ties $(r,s)$-regularity of $A$ to the quotient $N_A(H)/H$ being $(r,s)$-regular in $N_G(H)/H$, linking subgroup regularity to quotient-regularity and to perfect codes via corollaries. The results extend prior subgroup-perfect-code criteria to $(r,s)$-regular sets, provide counterexamples to naive converses, and unify previous findings under a coset-graph framework, advancing the study of perfect codes in vertex-transitive graphs beyond Cayley and distance-transitive cases.
Abstract
A subset \( C \) of the vertex set \( V \) of a graph \( Γ= (V,E) \) is termed an $(r,s)$-regular set if each vertex in \( C \) is adjacent to exactly \( r \) other vertices in \( C \), while each vertex not in \( C \) is adjacent to precisely \( s \) vertices in \( C \). A specific case, known as a $(0,1)$-regular set, is referred to as a perfect code. In this paper, we will delve into $(r,s)$-regular sets in the context of vertex-transitive graphs. It is noteworthy that any vertex-transitive graph can be represented as a coset graph \( \Cos(G,H,U) \). When examining a group \( G \) and a subgroup \( H \) of \( G \), a subgroup \( A \) that encompasses \( H \) is identified as an $(r,s)$-regular set related to the pair \( (G,H) \) if there exists a coset graph \( \Cos(G,H,U) \) such that the set of left cosets of \( H \) in \( A \) forms an $(r,s)$-regular set within this graph. In this paper, we present both a necessary and sufficient condition for determining when a normal subgroup \( A \) that includes \( H \) as a normal subgroup qualifies as an $(r,s)$-regular set for the pair \( (G,H) \). Furthermore, if \( A \) is a normal subgroup of \( G \) containing \( H \), we establish a relationship between \( A \) being a perfect code of \( (G,H) \) and the quotient \( N_A(H)/H \) being a perfect code of \(( N_G(H)/H, {1_{N_{G}(H)/H}}) \).