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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.19991 |
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| _version_ | 1866917625364217856 |
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| author | Mafunda, Sonwabile Merzel, Jonathan L. Perry, K. E. Varvak, Anna |
| author_facet | Mafunda, Sonwabile Merzel, Jonathan L. Perry, K. E. Varvak, Anna |
| contents | We determine the paint cost spectrum for perfect $k$-ary trees.
A coloring of the vertices of a graph $G$ with $d$ colors is said to be \emph{$d$-distinguishing} if only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number of $G$ and is denoted $\mbox{dist}(G).$ The \emph{paint cost of $d$-distinguishing $G$}, denoted $ρ^d(G)$, is the minimum size of the complement of a color class over all $d$-distinguishing colorings. A subset $S$ of the vertices of $G$ is said to be a \emph{fixing set} for $G$ if the only automorphsim that fixes the vertices in $S$ pointwise is the trivial automorphism. The cardinality of a smallest fixing set is denoted $\mbox{fix}(G)$. In this paper, we explore the breaking of symmetry in perfect $k$-ary trees by investigating what we define as the \emph{paint cost spectrum} of a graph $G$: $(\mbox{dist}(G); ρ^{\mbox{dist}(G)}(G), ρ^{\mbox{dist}(G)+1}(G), \dots, ρ^{\mbox{fix}(G)+1}(G))$ and the \emph{paint cost ratio} of $G$, which is defined to be the fraction of paint costs in the paint cost spectrum equal to $\mbox{fix}(G)$. We determine both the paint cost spectrum and the paint cost ratio completely for perfect $k$-ary trees.
We also prove a lemma that is of interest in its own right: given an $n$-tuple, $n \geq 2$ of distinct elements of an ordered abelian group and $1 \leq k \leq n! -1$, there exists a $k \times n$ row permuted matrix with distinct column sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Paint cost spectrum of perfect $k$-ary trees Mafunda, Sonwabile Merzel, Jonathan L. Perry, K. E. Varvak, Anna Combinatorics 05C25 We determine the paint cost spectrum for perfect $k$-ary trees. A coloring of the vertices of a graph $G$ with $d$ colors is said to be \emph{$d$-distinguishing} if only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number of $G$ and is denoted $\mbox{dist}(G).$ The \emph{paint cost of $d$-distinguishing $G$}, denoted $ρ^d(G)$, is the minimum size of the complement of a color class over all $d$-distinguishing colorings. A subset $S$ of the vertices of $G$ is said to be a \emph{fixing set} for $G$ if the only automorphsim that fixes the vertices in $S$ pointwise is the trivial automorphism. The cardinality of a smallest fixing set is denoted $\mbox{fix}(G)$. In this paper, we explore the breaking of symmetry in perfect $k$-ary trees by investigating what we define as the \emph{paint cost spectrum} of a graph $G$: $(\mbox{dist}(G); ρ^{\mbox{dist}(G)}(G), ρ^{\mbox{dist}(G)+1}(G), \dots, ρ^{\mbox{fix}(G)+1}(G))$ and the \emph{paint cost ratio} of $G$, which is defined to be the fraction of paint costs in the paint cost spectrum equal to $\mbox{fix}(G)$. We determine both the paint cost spectrum and the paint cost ratio completely for perfect $k$-ary trees. We also prove a lemma that is of interest in its own right: given an $n$-tuple, $n \geq 2$ of distinct elements of an ordered abelian group and $1 \leq k \leq n! -1$, there exists a $k \times n$ row permuted matrix with distinct column sums. |
| title | Paint cost spectrum of perfect $k$-ary trees |
| topic | Combinatorics 05C25 |
| url | https://arxiv.org/abs/2403.19991 |