On Distinguishing Graphs and Cost Number using Automorphism Representations

TL;DR

This paper investigates the relationship between the cost of distinguishing colorings and the determining number of graphs, introducing automorphism representations to analyze when these parameters can differ significantly.

Contribution

It solves an open problem for graphs with determining number 2 by showing the cost is bounded and small, and develops automorphism representations to classify graphs by their automorphism structure.

Findings

For Det(G)=2, the cost ρ(G) is bounded and takes small values 2, 3, or 4.

Automorphism representations determine when graphs have the same distinguishing properties.

Graphs with equivalent automorphism representations share the same distinguishing number.

Abstract

A distinguishing coloring of a graph is a vertex coloring such that only the identity automorphism of the graph preserves the coloring. A 2-distinguishable graph is a graph which can be distinguished using 2 colors. The cost $\rho(G)$ of a 2-distinguishable graph is the smallest size of a color set of a distinguishing coloring of $G$. The determining number of a graph, $Det(G)$, is the minimum number of nodes, which if fixed by a coloring, would ensure that the coloring distinguishes the entire graph. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) posed an open problem which asks if $\rho(G)$ and $Det(G)$ can be arbitrarily far apart. It is trivial that it cannot be so for the case $Det(G) = 1$ but the answer was unknown for $Det(G) \geq 2$. We solve this problem for the case $Det(G) = 2$. We show that for the case $Det(G) = 2$, that not only is the cost bounded but in fact it takes small values with $\rho(G) = 2, \ 3$ or $4$. In order to establish this, the concept of the automorphism representation of a graph is developed. Graphs having equivalent automorphism representations implies that they have the same distinguishing number (note that just having isomorphic automorphism groups is not enough for this to hold). This prompts a factoring of graphs by which two graphs are distinguishably equivalent iff they have equivalent automorphism representations.