The List Distinguishing Number Equals the Distinguishing Number for Interval Graphs

A $\mathit{\text{distinguishing}}\mathit{\text{coloring}}$ of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The $\mathit{\text{distinguishing}}\mathit{\text{number}}$ of $G$ is the minimum $k$ such that $G$ has a distinguishing coloring where each vertex is assigned a color from $\{1,\dots ,k\}$. A $\mathit{\text{list}}\mathit{\text{assignment}}$ to $G$ is an assignment $L=\{L(v){\}}_{v\in V(G)}$ of lists of colors to the vertices of $G$. A $\mathit{\text{distinguishing}}L\mathit{\text{-coloring}}$ of $G$ is a distinguishing coloring of $G$ where the color of each vertex $v$ comes from $L(v)$. The {\it list distinguishing number} of $G$ is the minimum $k$ such that every list assignment to $G$ in which $\mathrm{\mid }L(v)\mathrm{\mid }=k$ for all $v\in V(G)$ yields a distinguishing $L$-coloring of $G$. We prove that if $G$ is an interval graph, then its distinguishing number and list distinguishing number are equal.