Dichotomies properties on computational complexity of S-packing coloring problems

Abstract

This work establishes the complexity class of several instances of the S-packing coloring problem: for a graph G, a positive integer k and a nondecreasing list of integers S = (s1, . . . , s k ), G is S-colorable if its vertices can be partitioned into sets Si, i = 1, . . . , k, where each Si is an si-packing (a set of vertices at pairwise distance greater than si). In particular we prove a dichotomy between NP-complete problems and polynomial-time solvable problems for lists of at most four integers.Let S k d be a list only containing k integers d. The problem S k 1 -COL corresponds to the k-coloring problem which is known to be NP-complete for k ≥ 3. The S-coloring generalizes coloring with distance constraints like the packing coloring or the distance coloring of a graph. We denote by P-COL, the problem (1, 2, . . . , k)-COL for a graph G and an integer k (with G and k as input). The packing chromatic number [9] of G is the least integer k such that G is (1, 2, . . . , k)-colorable. A series of works [4,6,8,9] considered the packing chromatic number of infinite grids. The d-distance chromatic number [13] of G is the least integer k such that G is S k d -colorable. Initially, the concept of S-coloring has been introduced by Goddard et al. [9] and Fiala et al. [7]. The S-coloring problem was considered in other papers [10,11].The S-coloring problem, with |S| = 3 has been introduced by Goddard et al. [9] in order to determine the complexity of the packing chromatic number when k = 4. Moreover, Goddard and Xu [10] have proven that for |S| = 3, S-COL is NP-complete if s 1 = s 2 = 1 or if s 1 = 1 and s 2 = s 3 = 2 and polynomial-time solvable otherwise. About the complexity of S-COL, Fiala et al. [7] have proven that P-COL is NP-complete for trees and Argiroffo et al. [1,2] have proven that P-COL is polynomial-time solvable on some classes of graphs.In the second section, for a list S of three integers, we determine the family of S-colorable trees. Moreover, we determine dichotomies on cubic graphs, subcubic graphs and bipartite graphs. In the third section, we determine polynomialtime solvable and NP-complete instances of S-COL, for unfixed size of lists. We use these results to determine a dichotomy between NP-complete instances and polynomial-time solvable instances of S-COL for |S| ≤ 4.Note that for any nondecreasing list of integers S, we have S-COL in NP.

A survey on packing colorings

“…Gastineau [29] strengthened Theorem 7.49 by considering subcubic graphs, cubic graphs, bipartite graphs, and trees. His findings are summarized in Table 5, where S-Packing Coloring is abbreviated as S-PC.…”

“…If G is any graph that admits an S-packing coloring c and c(v) = 1 for some v ∈ V (G), then deg(v) ≤ k − 1 since no pair of neighbors of v are assigned the same color under c. An immediate consequence of this is that no cubic graph is (1, 2, 2)-packing colorable. On the other hand, there exist subcubic graphs that are (1, 2, 2)-packing colorable (for example, any path), and Gastineau [29] proved that it is NP-complete to determine if a subcubic bipartite graph is (1, 2, 2)-packing colorable. However, by allowing a partition of V (G) into an independent set and at least six 2-packings we have the following result.…”

“…29 [14, Theorem 3.2]. If G is a generalized prism of a cycle, then G is(1, 1, 2, 2)-packing colorable if and only if G is not the Petersen graph.…”

Spectrum graph coloring to improve Wi-Fi channel assignment in a real-world scenario via edge contraction

“…, p k )-labellings [14], where two vertices separated by a distance i must be labelled with colors whose mutual distance is at least p i . Some coloring problems have also arisen in direct connection with the frequency assignment problem, such as T -colorings [24] and the S-packing coloring [10]. In the former, given a set T of non-negative integers that represent disallowed separations between channels, the difference between two colors of adjacent vertices must not belong to T .…”

Packing Chromatic Number of Subdivisions of Cubic Graphs

“…The concept has attracted a considerable attention recently: there are around 30 papers on the topic (see e.g. [1,3,4,5,6,7,8,9,10,11,12,13,22] and references in them). In particular, Fiala and Golovach [10] proved that finding the packing chromatic number of a graph is NP-hard even in the class of trees.…”