Packing colorings of subcubic outerplanar graphs

Abstract

Given a graph G and a nondecreasing sequence S = (s1, . . . , s k ) of positive integers, the mapping c : V (G) −→ {1, . . . , k} is called an Spacking coloring of G if for any two distinct vertices x and y in c −1 (i), the distance between x and y is greater than si. The smallest integer k such that there exists a (1, 2, . . . , k)-packing coloring of a graph G is called the packing chromatic number of G, denoted χρ(G). The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs.In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by 7. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a (1, 2, 2, 2)-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a (1, 2, 2, 2)-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a (1, 2, 2, 3)-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an Spacking coloring for S = (1, 3, . . . , 3), where 3 appears ∆ times (∆ being the maximum degree of vertices), and if one of the integers 3 is replaced by 4 in the sequence S. Also, there exist outerplanar bipartite graphs that do not admit an S-packing coloring.

A survey on packing colorings

“…Brešar, Gastineau, and Togni [10] provided some additional results about S-packing colorings of outerplanar graphs. Theorem 7.40 is complemented by two examples showing that the theorem is best possible for subcubic outerplanar graphs.…”

“…,3,4), where 3 appears ∆(G) − 1 times, there exists a bipartite outerplanar graph that does not admit an S-packing coloring. For subcubic outerplanar graphs that have no triangles, they proved the following result.Theorem 7.40[10, Theorem 3]. If G is a subcubic, triangle-free, outerplanar graph, then G is (1, 2, 2, 2)-packing colorable.…”

A characterization of 4-$χ_S$-vertex-critical graphs for $S\succeq (1,3,3,3,\ldots)$

“…The development on the packing chromatic number up to 2020 has been summarized in the substantial survey [6]. Research into this concept is still flourishing, the developments after the survey include [1,2,5,8,10].…”

“…If S 1 = (s 1 1 , s and G admits an S 2 -packing k-coloring, then, clearly, G also admits an S 1 -packing k-coloring. In [11,Theorem 3.1], Gastineau proved the following appealing dichotomy result: If S is a packing sequence with |S| = 4, then the decision problem whether a given graph G admits an S-packing coloring is polynomial-time solvable if S S ′ , where S ′ ∈ { (2,3,3,3), (2,2,3,4), (1,4,4,4), (1,2,5,6)}, and NP-complete otherwise.…”

Packing $(1,1,2,2)$-coloring of some subcubic graphs

“…Many similar colorings have also been considered (e.g. [3,11,16,18,20,21]). In particular, Gastineau and Togni [16] showed subcubic graphs are packing (1, 2, 2, 2, 2, 2, 2)-colorable and packing (1, 1, 2, 2, 2)-colorable.…”