Derangement Representation of Graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913322617536512 |
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| author | Ashofteh, Somayeh Iradmusa, Moharram N. |
| author_facet | Ashofteh, Somayeh Iradmusa, Moharram N. |
| contents | A derangement $k$-representation of a graph $G$ is a map $π$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v $ and $u$ are adjacent if and only if $π(v)(i) \neq π(u)(i)$ for each $i \in \{1,2,3,\ldots,k\}$. The derangement representation number of $G$ denoted by $drn(G)$, is the minimum of $k$ such that $G$ has a derangement $k$-representation. In this paper, we prove that any graph has a derangement $k$-representation. Also, we obtain some lower and upper bounds for $drn(G)$, in terms of the basic parameters of $G$. Finally, we determine the exact value or give the better bounds of the derangement representation number of some classes of graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13424 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derangement Representation of Graphs Ashofteh, Somayeh Iradmusa, Moharram N. Combinatorics A derangement $k$-representation of a graph $G$ is a map $π$ of $V(G)$ to the symmetric group $S_k$, such that for any two vertices $v$ and $u$ of $V(G)$, $v $ and $u$ are adjacent if and only if $π(v)(i) \neq π(u)(i)$ for each $i \in \{1,2,3,\ldots,k\}$. The derangement representation number of $G$ denoted by $drn(G)$, is the minimum of $k$ such that $G$ has a derangement $k$-representation. In this paper, we prove that any graph has a derangement $k$-representation. Also, we obtain some lower and upper bounds for $drn(G)$, in terms of the basic parameters of $G$. Finally, we determine the exact value or give the better bounds of the derangement representation number of some classes of graphs. |
| title | Derangement Representation of Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.13424 |