Paint cost spectrum of perfect $k$-ary trees

Abstract

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.

Figures (3)

• Figure 1: 3-distinguishing coloring of a perfect trinary tree.
• Figure 2: Frugal coloring of depth-3 trinary perfect tree, using five colors: white (neutral), black, green, cyan, and red.
• Figure 3: One of many efficient 4-colorings of $T^3_3$.

Theorems & Definitions (25)

• Theorem 1
• proof
• Proposition 1
• proof
• Lemma 1
• proof
• Example 4.1
• Lemma 2
• proof
• Theorem 2