9 research outputs found
Perfect codes in 2-valent Cayley digraphs on abelian groups
For a digraph , a subset of is a perfect code if
is a dominating set such that every vertex of is dominated by exactly
one vertex in . In this paper, we classify strongly connected 2-valent
Cayley digraphs on abelian groups admitting a perfect code, and determine
completely all perfect codes of such digraphs
On subgroup perfect codes in Cayley sum graphs
A perfect code in a graph is an independent set of vertices of
such that every vertex outside of is adjacent to a unique vertex
in , and a total perfect code in is a set of vertices of
such that every vertex of is adjacent to a unique vertex in
. Let be a finite group and a normal subset of . The Cayley sum
graph of with the connection set is the graph with
vertex set and two vertices and being adjacent if and only if
and . In this paper, we give some necessary conditions of a
subgroup of a given group being a (total) perfect code in a Cayley sum graph of
the group. As applications, the Cayley sum graphs of some families of groups
which admit a subgroup as a (total) perfect code are classified
Perfect codes in quintic Cayley graphs on abelian groups
A subset of the vertex set of a graph is called a perfect code
of if every vertex of is at distance no more than one to
exactly one vertex in . In this paper, we classify all connected quintic
Cayley graphs on abelian groups that admit a perfect code, and determine
completely all perfect codes of such graphs