Perfect codes in circulant graphs

Abstract

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such that every vertex of $V$ is adjacent to exactly one vertex in $C$. A perfect code in the Hamming graph $H(n, q)$ agrees with a $q$-ary perfect 1-code of length $n$ in the classical setting. In this paper we give a necessary and sufficient condition for a circulant graph of degree $p-1$ to admit a perfect code, where $p$ is an odd prime. We also obtain a necessary and sufficient condition for a circulant graph of order $n$ and degree $p^l-1$ to have a perfect code, where $p$ is a prime and $p^l$ the largest power of $p$ dividing $n$. Similar results for total perfect codes are also obtained in the paper

Subgroup perfect codes in Cayley sum graphs

“…For the past few years, perfect codes in Cayley graphs have attracted considerable attention, see, for example, [11,27,28]. In [14], Huang, Xia and Zhou first introduced the concept of a perfect code of a group G. A subset C of G is said to be a perfect code of G if C is a perfect code of some Cayley graph of G. In particular, a subgroup is said to be a subgroup perfect code of G if the subgroup is also a perfect code of G. Also in [14], they gave a necessary and sufficient condition for a normal subgroup of a group G to be a subgroup perfect code of G, and determined all the subgroup perfect codes of dihedral groups and some abelian groups.…”

Efficient domination in Cayley graphs of generalized dihedral groups

“…However, no general result in this direction has been obtained thus far which probably makes the problem, proposed in [17], of characterizing the vertex-transitive graphs admitting an efficient dominating set way too difficult to solve in general. In fact, even for the nicest possible situation in which the graph in question admits a regular action of a cyclic group (such graphs are called circulants) only a few partial results have been obtained thus far (see [6,9,10,13,18,21]). To better understand the situation, it is reasonable to study particular classes of vertex-transitive graphs.…”

Classification of Efficient Total Domination Sets of Circulant Graphs of Degree 5

“…In [11], the authors and Chen classified all circulant graphs of degree 4 containing an efficient total dominating set. In [12], Feng et al obtained a necessary and sufficient condition for the existence of efficient total dominating set of a circulant graph whose degree is an odd prime p.…”