A generalization of perfect codes in the presence of star multiset transpositions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916296125317120 |
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| author | Dejter, Italo J. |
| author_facet | Dejter, Italo J. |
| contents | Let $0<\ell\in\mathbb{Z}$. The notion of an efficient dominating set or perfect code $S$ of a graph $G$ is generalized to that of an efficient dominating$\,^\ell$-set or perfect$^\ell$code, of the graph $G$, meaning that each vertex $v$ of $V(G)\setminus S$ has exactly $\ell$ neighbors in $S$, instead of just one neighbor. Such generalization is applied to star $j$-set transposition graphs based on permutations of multisets with each symbol repeated $j$ times, ($j\in\{\ell,\ell-1\}$). In such vertex-transitive graphs this approach produces total colorings, efficient dominating sets, also called perfect codes, etc. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_04101 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalization of perfect codes in the presence of star multiset transpositions Dejter, Italo J. Combinatorics 05C62, 05C69, 015C70, 05C75, 94B25 Let $0<\ell\in\mathbb{Z}$. The notion of an efficient dominating set or perfect code $S$ of a graph $G$ is generalized to that of an efficient dominating$\,^\ell$-set or perfect$^\ell$code, of the graph $G$, meaning that each vertex $v$ of $V(G)\setminus S$ has exactly $\ell$ neighbors in $S$, instead of just one neighbor. Such generalization is applied to star $j$-set transposition graphs based on permutations of multisets with each symbol repeated $j$ times, ($j\in\{\ell,\ell-1\}$). In such vertex-transitive graphs this approach produces total colorings, efficient dominating sets, also called perfect codes, etc. |
| title | A generalization of perfect codes in the presence of star multiset transpositions |
| topic | Combinatorics 05C62, 05C69, 015C70, 05C75, 94B25 |
| url | https://arxiv.org/abs/2401.04101 |