### A survey on packing colorings

Boštjan Brešar, Jasmina Ferme, Sandi Klavžar, Douglas F. Rall
2020 Discussiones Mathematicae Graph Theory
If S = (a 1 , a 2 , . . .) is a non-decreasing sequence of positive integers, then an S-packing coloring of a graph G is a partition of V (G) into sets X 1 , X 2 , . . . such that for each pair of distinct vertices in the set X i , the distance between them is larger than a i . If there exists an integer k such that V (G) = X 1 ∪ · · · ∪ X k , then the partition is called an S-packing k-coloring. The S-packing chromatic number of G is the smallest k such that G admits an S-packing k-coloring.
more » ... cking k-coloring. If a i = i for every i, then the terminology reduces to packing colorings and packing chromatic number. Since the introduction of these generalizations of the chromatic number in 2008 more than fifty papers followed. Here we survey the state of the art on the packing coloring, and its generalization, the S-packing coloring. We also list several conjectures and open problems. A Survey on Packing Colorings 3 packing coloring and packing chromatic number were (and still are) used, the terms being coined by Brešar, Klavžar, and Rall . Because of the numerous papers written on the packing chromatic number and since it is of continuing interest, we think now is the time to collect results and open problems in a survey paper. In the next section we state some basic properties of the packing chromatic number; relate it with invariants α, β, χ, and ω; give partial results on trees; and present known complexity results. In Section 3, we consider one of the most exciting topics related to the packing chromatic number, namely, how it behaves on subcubic graphs. In the subsequent section, the known effects on the packing chromatic number of local operations (vertex deletion, edge deletion, edge contraction, edge subdivision) are summarized. In Section 5 the packing chromatic number of specific classes of graphs is given with an emphasis on Cartesian products and Sierpiński-type graphs. The packing chromatic number has also been studied on infinite graphs, which is the focus of Section 6. The main roles here are played by several families of lattices and of distance graphs. Section 7 presents what is known about S-packing colorings (for S = (1, 2, . . .)), which is a wide generalization of packing colorings. We conclude with a list of conjectures and open problems. In the rest of the introduction we collect definitions and notation needed throughout the paper. Given a graph G = (V (G), E(G)) and a positive integer i, an i-packing in G is a subset X of V (G) such that the distance d G (u, v) between any two distinct vertices u, v ∈ X is greater than i. (The distance d G (u, v) is the length of a shortest path between u and v in G.) The packing chromatic number χ ρ (G) of G is the smallest integer k such that the vertex set of G can be partitioned into sets X 1 , . . . , X k , where X i is an i-packing for each i ∈ [k] = {1, . . . , k}. Such a partition corresponds to a mapping c : V (G) → [k] such that X i = {u ∈ V (G) : c(u) = i}. This mapping has the property that c(u) = c(v) = i implies d G (u, v) > i; c is called a packing k-coloring. Along the way it will be useful to have the following generalization of the packing chromatic number in hand. Let S = (a 1 , a 2 , . . .) be an infinite, nondecreasing sequence of positive integers. (All sequences of positive integers in this paper, whether finite or infinite, are assumed to be non-decreasing.) An S-packing coloring of G is a mapping c : V (G) → N such that c −1 (i) is an a ipacking for each i ∈ N. When such a mapping c exists for a graph G, we say G is S-packing colorable. If k is a positive integer and c(u) ∈ [k] for each vertex u, then c is called an S-packing k-coloring or an (a 1 , . . . , a k )-packing coloring of G. The S-packing chromatic number of G, denoted χ S (G), is the smallest k such that G has an S-packing k-coloring. If G does not admit an S-packing k-coloring for any k ∈ N, then we set χ S (G) = ∞. For example, for any sequence S we have χ S (K N ) = ∞, where K N is the countably infinite complete graph. Note that