Paint Cost and the Frugal Distinguishing Number

TL;DR

This paper introduces new parameters related to graph distinguishing costs, defines the frugal distinguishing number, and computes these parameters for book graphs, highlighting their potential to differ significantly from traditional distinguishing numbers.

Contribution

The paper formally defines the paint cost parameters for graph distinguishing, introduces the frugal distinguishing number, and computes these for specific graph families, expanding understanding of graph symmetry breaking costs.

Findings

Upper and lower paint costs can be arbitrarily large multiples of each other.

The frugal distinguishing number for book graphs is explicitly determined.

Paint costs for book graphs depend on parameters m and n, with specific formulas provided.

Abstract

You are handed a graph with vertices in a neutral color and asked to color a subset of vertices with expensive paints in $d$ colors in such a way that only the trivial symmetry preserves the color classes. Your goal is to minimize the number of vertices needing this expensive paint. This paper address the issues surrounding your choices. In particular, a graph is said to be $d$-distinguishable if there exists a coloring with $d$ colors so that only the trivial automorphism preserves the color classes. The distinguishing number of $G$, denoted ${\rm Dist}(G)$, is the smallest $d$ for which $G$ is $d$-distinguishable. We define the -paint cost of $d$-distinguishing, denoted $\rho^d(G)$, to be the minimum number of vertices that need to be painted to $d$-distinguish $G$. This cost varies with $d$. The maximum paint cost for $G$ is called the upper paint cost, denoted $\rho^u(G)$, and occurs when $d={\rm Dist}(G)$; the minimum paint cost is called the lower paint cost, denoted $\rho^\ell(G)$. Further, we define the smallest $d$ for which the paint cost is $\rho^\ell(G)$, to be the frugal distinguishing number, ${\rm Fdist}(G)$. In this paper we formally define $\rho^d(G)$, $\rho^u(G)$, $\rho^\ell(G)$, and ${\rm Fdist}(G)$. We also show that $\rho^u(G)$ and $\rho^\ell(G)$, as well as ${\rm Fdist}(G)$ and ${\rm Dist}(G)$, can be arbitrarily large multiples of each other. Lastly, we find these parameters for the book graph $B_{m,n}$, summarized as follows. For $n\geq 2$ and $m\geq 4$, we show $\bullet$ $\rho^\ell(B_{m,n}) = n-1;$ $\bullet$ $\rho^u(B_{m,n}) \geq (m-2) \left( n-k^{m-3} \right) +1$, where $k={\rm Dist}(B_{m,n});$ $\bullet$ ${\rm Fdist}(B_{m,n}) = 2+\left\lfloor \frac{n-1}{m-2} \right\rfloor.$