Given a group $Gamma$ acting on a set $X$, a $k$-coloring $phi:Xto{1,dots,k}$ of $X$ is distinguishing with respect to $Gamma$ if the only $gammain Gamma$ that fixes $phi$ is the identity action. The distinguishing number of the action $Gamma$, denoted $D_{Gamma}(X)$, is then the smallest positive integer $k$ such that there is a distinguishing $k$-coloring of $X$ with respect to $Gamma$. This notion has been studied in a number of settings, but by far the largest body of work has been concerned with finding the distinguishing number of the action of the automorphism group of a graph $G$ upon its vertex set, which is referred to as the distinguishing number of $G$. The distinguishing number of a group action is a measure of how difficult it is to break all of the permutations arising from that action. In this paper, we aim to further differentiate the resilience of group actions with the same distinguishing number. In particular, we introduce a precoloring extension framework to address this issue. A set $S subseteq X$ is a fixing set for $Gamma$ if for every non-identity element $gamma in Gamma$ there is an element $s in S$ such that $gamma(s) eq s$. The distinguishing extension number $operatorname{ext}_D(X,Gamma;k)$ is the minimum number $m$ such that for all fixing sets $W subseteq X$ with $|W| geq m$, every $k$-coloring $c : X setminus W to [k]$ can be extended to a $k$-coloring that distinguishes $X$. In this paper, we prove that $operatorname{ext}_D(mathbb{R},operatorname{Aut}(mathbb{R}),2) =4$, where $operatorname{Aut}(mathbb{R})$ is comprised of compositions of translations and reflections. We also consider the distinguishing extension number of the circle and (finite) cycles, obtaining several exact results and bounds.
تحميل البحث