Conflict-Free Vertex Coloring Of Planar Graphs (original) (raw)

Conflict-Free Coloring of Graphs

SIAM Journal on Discrete Mathematics

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry and are well studied in graph theory. Here we study the natural problem of the conflict-free chromatic number \chi CF (G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N [v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N (v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K k+1 as a minor, then \chi CF (G) \leq k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k \in \{ 1, 2, 3\} , it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors.

A novel heuristic for the coloring of planar graphs

2018

A novel algorithm is proposed for the coloring of planar graphs based on the construction of a maximal independent set S. The maximal independent set S must fullfil certain characteristics. It must contain the vertex that appears in the maximum number of odd cycles of G. The construction of S considers the internal-face graph of the input graph G in order to select the vertices that decomposes a maximal number of initial faces of G. The traversing in pre-order of the internal-face graph Gf , of the planar graph G, provides us of a strategy in the construction of the maximal independent set S that reduces (G− S) in a polygonal tree that will be 3-colorable.

A Heuristic for the Coloring of Planar Graphs

Electronic Notes in Theoretical Computer Science, 2020

We present an algorithm for the coloring of planar graphs based on the construction of a maximal independent set S of the input graph. The maximal independent set S must fulfill certain characteristics. For example, S contains the vertex that appears in a maximum number of odd cycles of G. The construction of S considers the internal-face graph of the input graph G in order to select each vertex belonging to a maximal number of odd faces of G. The traversing in pre-order on the internal-face graph Gf of the input planar graph G provides us of a strategy for the construction of partial maximal independent sets of critical regions of Gf. Thus, the union of these partial maximal independent sets forms a maximal independent set S of G. This allows us to color first the vertices that are crucial for decomposing G in a graph (G − S), which is a polygonal tree, and therefore, is 3-colorable.

Building a Maximal Independent Set for the Vertex-coloring Problem on Planar Graphs

Electronic Notes in Theoretical Computer Science, 2020

We analyze the vertex-coloring problem restricted to planar graphs and propose to consider classic wheels and polyhedral wheels as basic patterns for the planar graphs. We analyze the colorability of the composition among wheels and introduce a novel algorithm based on three rules for the vertex-coloring problem. These rules are: 1) Selecting vertices in the frontier. 2) Processing subsumed wheels. 3) Processing centers of the remaining wheels. Our method forms a maximal independent set S 1 ⊂ V (G) consisting of wheel's centers, and a maximum number of vertices in the frontier of the planar graph. Thus, we show that if the resulting graph G = (G − S 1) is 3-colorable, then this implies the existence of a valid 4-coloring for G.

New Linear-Time Algorithms for Edge-Coloring Planar Graphs

Algorithmica, 2007

We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree ∆ with max{∆, 9} colors. Thus the coloring is optimal for graphs with maximum degree ∆ ≥ 9. Moreover for ∆ = 4, 5, 6 we give linear-time algorithms that use ∆ + 2 colors. These results improve over the algorithms of Chrobak and Yung [1] and of Chrobak and Nishizeki [2] which color planar graphs using max{∆, 19} colors in linear time or using max{∆, 9} colors in O(n log n) time.

An exact approach for the Vertex Coloring Problem

Discrete Optimization, 2011

Given an undirected graph G = (V , E), the Vertex Coloring Problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. In this paper, we present an exact algorithm for the solution of VCP based on the well-known Set Covering formulation of the problem. We propose a Branch-and-Price algorithm embedding an effective heuristic from the literature and some methods for the solution of the slave problem, as well as two alternative branching schemes. Computational experiments on instances from the literature show the effectiveness of the algorithm, which is able to solve, for the first time to proven optimality, five of the benchmark instances in the literature, and reduce the optimality gap of many others.

Remarks on proper conflict-free colorings of graphs

Discrete Mathematics

A vertex coloring of a graph is said to be conflict-free with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods, arXiv preprint], the minimum number of colors in any such proper coloring of graph G is the PCF chromatic number of G, denoted χ pcf (G). In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on χ pcf (G) in terms of other graph parameters. In particular, we show that χ pcf (G) ≤ 5∆(G) 2 and characterize equality. Several sufficient conditions for PCF k-colorability of graphs are established for 4 ≤ k ≤ 6. The paper concludes with few open problems.

A survey on vertex coloring problems

International Transactions in Operational Research, 2010

This paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms proposed in the literature are reported. The second part of the paper is devoted to some generalizations of the problem, which are obtained by considering additional constraints [Bandwidth (Multi) Coloring Problem, Bounded Vertex Coloring Problem] or an objective function with a special structure (Weighted Vertex Coloring Problem). The extension of the models for the classical VCP to the considered problems and the best performing algorithms from the literature, as well as the corresponding computational results, are reported.

A linear 5-coloring algorithm of planar graphs

Journal of Algorithms, 1981

A simple linear algorithm is presented for coloring planar graphs with at most five colors. The algorithm employs a recursive reduction of a graph involving the deletion of a vertex of degree 6 or less possibly together with the identification of its several neighbors.