TY - JOUR
T1 - Distinguishing density and the Distinct Spheres Condition
AU - Imrich, Wilfried
AU - Lehner, Florian
AU - Smith, Simon M.
PY - 2020/10
Y1 - 2020/10
N2 - If a graph G has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of G fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of G, and the parts are called its color classes. If G admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain sense. In this case we say that G has 2-distinguishing density zero. An extreme example of this would be an infinite graph admitting a 2-distinguishing coloring in which one of the color classes is finite. The Infinite Motion Conjecture is a well-known open conjecture about 2-distinguishability. A graph G is said to have infinite motion if every nontrivial automorphism of G moves infinitely many vertices, and the conjecture states that every connected, locally finite graph with infinite motion is 2-distinguishable. In this paper we show that for many classes of graphs for which the Infinite Motion Conjecture is known to hold, the graphs have 2-distinguishing density zero.
AB - If a graph G has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of G fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of G, and the parts are called its color classes. If G admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain sense. In this case we say that G has 2-distinguishing density zero. An extreme example of this would be an infinite graph admitting a 2-distinguishing coloring in which one of the color classes is finite. The Infinite Motion Conjecture is a well-known open conjecture about 2-distinguishability. A graph G is said to have infinite motion if every nontrivial automorphism of G moves infinitely many vertices, and the conjecture states that every connected, locally finite graph with infinite motion is 2-distinguishable. In this paper we show that for many classes of graphs for which the Infinite Motion Conjecture is known to hold, the graphs have 2-distinguishing density zero.
UR - http://www.scopus.com/inward/record.url?scp=85083880136&partnerID=8YFLogxK
U2 - 10.1016/j.ejc.2020.103139
DO - 10.1016/j.ejc.2020.103139
M3 - Article
SN - 0195-6698
VL - 89.2020
JO - European journal of combinatorics
JF - European journal of combinatorics
IS - October
M1 - 103139
ER -