914 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
Some non-existence results for distance- ovoids in small generalized polygons
We give a computer-based proof for the non-existence of distance- ovoids
in the dual split Cayley hexagon .
Furthermore, we give upper bounds on partial distance- ovoids of
for .Comment: 10 page
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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