1,159 research outputs found

    Perfect domination in regular grid graphs

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    We show there is an uncountable number of parallel total perfect codes in the integer lattice graph Ξ›{\Lambda} of R2\R^2. In contrast, there is just one 1-perfect code in Ξ›{\Lambda} and one total perfect code in Ξ›{\Lambda} restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products CmΓ—CnC_m\times C_n with parallel total perfect codes, and the dd-perfect and total perfect code partitions of Ξ›{\Lambda} and CmΓ—CnC_m\times C_n, the former having as quotient graph the undirected Cayley graphs of Z2d2+2d+1\Z_{2d^2+2d+1} with generator set {1,2d2}\{1,2d^2\}. For r>1r>1, generalization for 1-perfect codes is provided in the integer lattice of Rr\R^r and in the products of rr cycles, with partition quotient graph K2r+1K_{2r+1} taken as the undirected Cayley graph of Z2r+1\Z_{2r+1} with generator set {1,...,r}\{1,...,r\}.Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi

    Perfect codes in quintic Cayley graphs on abelian groups

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    A subset CC of the vertex set of a graph Ξ“\Gamma is called a perfect code of Ξ“\Gamma if every vertex of Ξ“\Gamma is at distance no more than one to exactly one vertex in CC. 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

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    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

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    A perfect code CC in a graph Ξ“\Gamma is an independent set of vertices of Ξ“\Gamma such that every vertex outside of CC is adjacent to a unique vertex in CC, and a total perfect code CC in Ξ“\Gamma is a set of vertices of Ξ“\Gamma such that every vertex of Ξ“\Gamma is adjacent to a unique vertex in CC. Let GG be a finite group and XX a normal subset of GG. The Cayley sum graph CS(G,X)\mathrm{CS}(G,X) of GG with the connection set XX is the graph with vertex set GG and two vertices gg and hh being adjacent if and only if gh∈Xgh\in X and gβ‰ hg\neq h. 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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