Distinguishing Orthogonality Graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| _version_ | 1866929382268862464 |
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| author | Boutin, Debra Cockburn, Sally |
| author_facet | Boutin, Debra Cockburn, Sally |
| contents | A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertices with $d$ labels so that only the trivial automorphism preserves the labels. The smallest such $d$ is the distinguishing number, Dist($G$). A subset of vertices $S$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called the determining number, Det($G$). The orthogonality graph $Ω_{2k}$ has vertices which are bitstrings of length $2k$ with an edge between two vertices if they differ in precisely $k$ bits. This paper shows that Det($Ω_{2k}$) $= 2^{2k-1}$ and that if $\binom{m}{2} \geq 2k$ then $2<$ Dist($Ω_{2k}$) $\leq m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_00092 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Distinguishing Orthogonality Graphs Boutin, Debra Cockburn, Sally Combinatorics 05C15, 05E18 A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertices with $d$ labels so that only the trivial automorphism preserves the labels. The smallest such $d$ is the distinguishing number, Dist($G$). A subset of vertices $S$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called the determining number, Det($G$). The orthogonality graph $Ω_{2k}$ has vertices which are bitstrings of length $2k$ with an edge between two vertices if they differ in precisely $k$ bits. This paper shows that Det($Ω_{2k}$) $= 2^{2k-1}$ and that if $\binom{m}{2} \geq 2k$ then $2<$ Dist($Ω_{2k}$) $\leq m$. |
| title | Distinguishing Orthogonality Graphs |
| topic | Combinatorics 05C15, 05E18 |
| url | https://arxiv.org/abs/2001.00092 |