List-Distinguishing Colorings of Graphs

Abstract

A coloring of the vertices of a graph $G$ is said to be distinguishing provided that no nontrivial automorphism of $G$ preserves all of the vertex colors. The distinguishing number of $G$, denoted $D(G)$, is the minimum number of colors in a distinguishing coloring of $G$. The distinguishing number, first introduced by Albertson and Collins in 1996, has been widely studied and a number of interesting results exist throughout the literature. In this paper, the notion of distinguishing colorings is extended to that of list-distinguishing colorings. Given a family $L=\{L(v)\}_{v\in V(G)}$ of lists assigning available colors to the vertices of $G$, we say that $G$ is $L$-distinguishable if there is a distinguishing coloring $f$ of $G$ such that $f(v)\in L(v)$ for all $v$. The list-distinguishing number of $G$, $D_{\ell}(G)$, is the minimum integer $k$ such that $G$ is $L$-distinguishable for any assignment $L$ of lists with $|L(v)|=k$ for all $v$. Here, we determine the list-distinguishing number for several families of graphs and highlight a number of distinctions between the problems of distinguishing and list-distinguishing a graph.

The list distinguishing number of Kneser graphs

“…Our choices also ensure that the pallettes of vertices 1 and 3 are different, so it follows that σ fixes 1, 2, 3. Since c 46 = c 56 , σ = (45), (456), (465) and since c 14 , c 15 = c 16 , σ = (46), (56), so σ is the identity map on [6].…”

“…In this paper, we prove that D l (G) = D(G) when G is a Kneser graph.An interesting variant of the distinguishing number of a graph, due to Ferrara, Flesch, and Gethner [6] goes as follows. Given an assignment L = {L(v)} v∈V (G) of lists of available colors to vertices of G, we say that G is L−distinguishable if there is a distinguishing coloring f of G such that f (v) ∈ L(v) for all v. The list distinguishing number of G, D l (G) is the minimum integer k such that G is L−distinguishable for any list assignment L with |L(v)| = k for all v. The list distinguishing number has generated a bit of interest recently (see [6,7,8] for some relevant results) primarily due to the following conjecture that appears in [6]: For any graph G, D l (G) = D(G). The paper [6], in which this notion was introduced and the conjecture was made, proves the same for cycles of size at least 6, cartesian products of cycle, and for graphs whose automorphism group is the Dihedral group.…”

List distinguishing parameters of trees

“…While not explicitly introduced in [11], it is natural to consider a list analogue of the distinguishing chromatic number. We say that G is properly L-distinguishable if there is a distinguishing coloring f of G chosen from the lists such that f is also a proper coloring of G. The list distinguishing chromatic number χ D ℓ (G) of G is the minimum integer k such that G is properly L-distinguishable for any assignment L of lists with |L(v)| ≥ k for all v.…”

List-distinguishing Cartesian products of cliques

“…Question 1 is unanswered, but in subsequent years, there has been an accumulation of evidence suggesting that the negative answer to Question 1 is correct. This includes proofs that D(G) = D ℓ (G) when G has a dihedral automorphism group [13], is a forest [14], and when G is an interval graph [17].…”