1,159 research outputs found
Perfect domination in regular grid graphs
We show there is an uncountable number of parallel total perfect codes in the
integer lattice graph of . In contrast, there is just one
1-perfect code in and one total perfect code in
restricting to total perfect codes of rectangular grid graphs (yielding an
asymmetric, Penrose, tiling of the plane). We characterize all cycle products
with parallel total perfect codes, and the -perfect and
total perfect code partitions of and , the former
having as quotient graph the undirected Cayley graphs of with
generator set . For , generalization for 1-perfect codes is
provided in the integer lattice of and in the products of cycles,
with partition quotient graph taken as the undirected Cayley graph
of with generator set .Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi
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
On perfect codes in Cartesian products of graphs
AbstractAssuming the existence of a partition in perfect codes of the vertex set of a finite or infinite bipartite graph G we give the construction of a perfect code in the Cartesian product Gβ‘Gβ‘P2. Such a partition is easily obtained in the case of perfect codes in Abelian Cayley graphs and we give some examples of applications of this result and its generalizations
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
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