On fixing and distinguishing numbers of trees
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912981476966400 |
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| author | Buchanan, Calum Dankelmann, Peter Harris, Isabel Horn, Paul Perry, K. E. Rivett-Carnac, Emily |
| author_facet | Buchanan, Calum Dankelmann, Peter Harris, Isabel Horn, Paul Perry, K. E. Rivett-Carnac, Emily |
| contents | A graph $G$ is $D$-distinguishable if there is a labeling of its vertices with $D$ labels such that the only automorphism of $G$ which preserves the labeling is the identity. The distinguishing number of $G$ is the minimum value $D$ for which $G$ is $D$-distinguishable. The fixing number of $G$ is the minimum cardinality of a subset of the vertices of $G$ which is fixed pointwise only by the trivial automorphism. We prove that the fixing number of any $2$-distinguishable tree of order $n \geq 3$ is at most $4n/11$, or at most $(D-1)n / (D+1)$ for a $D$-distinguishable tree ($D \geq 3$). For every $D$ and $r$ at least $2$, we characterize the $D$-distinguishable trees with radius $r$ by constructing a universal tree $T_r^D$ which has the property that a tree $T$ of radius $r$ is $D$-distinguishable if and only if $T$ is a union of branches of $T_r^D$. We obtain a similar collection of universal trees for the property of having a constant paint cost spectrum, i.e., the minimum size of the complement of a color class in a distinguishing $D$-coloring of $T$ is equal to the fixing number. Finally, we prove bounds on the distinguishing and fixing numbers of a tree in terms of the eccentricities of its vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23820 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On fixing and distinguishing numbers of trees Buchanan, Calum Dankelmann, Peter Harris, Isabel Horn, Paul Perry, K. E. Rivett-Carnac, Emily Combinatorics 05C15, 05C25, 05C05 A graph $G$ is $D$-distinguishable if there is a labeling of its vertices with $D$ labels such that the only automorphism of $G$ which preserves the labeling is the identity. The distinguishing number of $G$ is the minimum value $D$ for which $G$ is $D$-distinguishable. The fixing number of $G$ is the minimum cardinality of a subset of the vertices of $G$ which is fixed pointwise only by the trivial automorphism. We prove that the fixing number of any $2$-distinguishable tree of order $n \geq 3$ is at most $4n/11$, or at most $(D-1)n / (D+1)$ for a $D$-distinguishable tree ($D \geq 3$). For every $D$ and $r$ at least $2$, we characterize the $D$-distinguishable trees with radius $r$ by constructing a universal tree $T_r^D$ which has the property that a tree $T$ of radius $r$ is $D$-distinguishable if and only if $T$ is a union of branches of $T_r^D$. We obtain a similar collection of universal trees for the property of having a constant paint cost spectrum, i.e., the minimum size of the complement of a color class in a distinguishing $D$-coloring of $T$ is equal to the fixing number. Finally, we prove bounds on the distinguishing and fixing numbers of a tree in terms of the eccentricities of its vertices. |
| title | On fixing and distinguishing numbers of trees |
| topic | Combinatorics 05C15, 05C25, 05C05 |
| url | https://arxiv.org/abs/2603.23820 |