The distinguishing number and distinguishing chromatic number for posets

Karen L. Collins, Ann N. Trenk

Published 2019-05-23Version 1

In this paper we introduce the distinguishing number of a poset $P$ as the minimum number of colors needed to color the points of $P$ so that any automorphism of $P$ preserves colors. We find the distinguishing number of any distributive lattice and certain classes of ranked planar posets by constructing appropriate colorings. In addition, we suggest two natural definitions for the distinguishing chromatic number of a poset. The first of these reduces to the width of the poset, but the second is more interesting and we prove an upper bound for distributive lattices.