Remarks on proper conflict-free colorings of graphs (original) (raw)
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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.
The concept of vertex coloring pose a number of challenging open problems in graph theory. Among several interesting parameters, the coloring parameter, namely the pseudoachromatic number of a graph stands a class apart. Although not studied very widely like other parameters in the graph coloring literature, it has started gaining prominence in recent years. The pseudoachromatic number of a simple graph G, denoted ψ(G), is the maximum number of colors used in a vertex coloring of G, where the adjacent vertices may or may not receive the same color but any two distinct pair of colors are represented by at least one edge in it. In this paper we have computed this parameter for a number of classes of graphs.
Conflict-Free Vertex Coloring Of Planar Graphs
2017
The conflict-free coloring problem is a variation of the vertex coloring problem, a classical NP-hard optimization problem. The conflict-free coloring problem aims to color a possibly proper subset of vertices such that there is a unique color within the closed neighborhood (a vertex and its neighbors) of every vertex. This paper presents recent findings and heuristics to solve the conflict-free coloring problem on both general graphs and planar graphs.
On the Local Colorings of Graphs
2008
A local coloring of a graph G is a function c : V (G) −→ N having the property that for each set S ⊆ V (G) with 2 ≤ |S| ≤ 3, there exist vertices u, v ∈ S such that |c(u) − c(v)| ≥ mS , where mS is the size of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ (c). The local chromatic number of G is χ (G) = min{χ (c)}, where the minimum is taken over all local colorings c of G. If χ (c) = χ (G), then c is called a minimum local coloring of G. The local coloring of graphs introduced by Chartrand et. al. in 2003. In this paper, following the study of this concept, first an upper bound for χ (G) where G is not complete graphs K4 and K5, is provided in terms of maximum degree Δ(G). Then the exact value of χ (G) for some special graphs G such as the cartesian product of cycles, paths and complete graphs is determined.
New results on generalized graph coloring
2004
For graph classes P 1 , . . . , P k , Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V 1 , . . . ,V k so that V j induces a graph in the class P j ( j = 1, 2, . . . , k). If P 1 = · · · = P k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k ≥ 3. Recently, this result has been generalized by showing that if all P i 's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs.
The maximum number of colorings of graphs of given order and size: A survey
Discrete Mathematics
Let m, n, λ be positive integers. What is the maximum number of proper vertex colorings in (at most) λ colors a graph with n vertices and m edges can have? On which graphs is this maximum attained? The question can be rephrased as the one of maximizing χ(G, λ), the the value of the chromatic polynomial of G at λ, over all graphs G with n vertices and m edges. This problem was stated independently by Wilf and Linial, and is still unsolved. In this article we survey the current state of the research directed at solving the problem.
On compact -edge-colorings: A polynomial time reduction from linear to cyclic
Discrete Optimization, 2011
A k-edge-coloring of a graph G = (V, E) is a function c that assigns an integer c(e) (called color) in {0, 1, · · · , k −1} to every edge e ∈ E so that adjacent edges get different colors. A k-edge-coloring is linear compact if the colors incident to every vertex are consecutive. The problem k − LCCP is to determine whether a given graph admits a linear compact k-edge coloring. A k-edge-coloring is cyclic compact if there are two positive integers av, bv in {0, 1, · · · , k − 1} for every vertex v such that the colors incident to v are exactly {av, (av + 1)mod k, · · · , bv}. The problem k − CCCP is to determine whether a given graph admits a cyclic compact k-edge coloring. We show that the k − LCCP with possibly imposed or forbidden colors on some edges is polynomially reducible to the k − CCCP when k ≥ 12, and to the 12 − CCCP when k < 12.
Maximum edge-colorings of graphs
Discussiones Mathematicae Graph Theory, 2015
An r-maximum k-edge-coloring of G is a k-edge-coloring of G having a property that for every vertex v of degree d G (v) = d, d ≥ r, the maximum color, that is present at vertex v, occurs at v exactly r times. The r-maximum index χ ′ r (G) is defined to be the minimum number k of colors needed for an r-maximum k-edge-coloring of graph G. In this paper we show that χ ′ r (G) ≤ 3 for any nontrivial connected graph G and r = 1 or 2. The bound 3 is tight. All graphs G with χ ′ 1 (G) = i, i = 1, 2, 3 are characterized. The precise value of the r-maximum index, r ≥ 1, is determined for trees and complete graphs.
Conflict-Free Colouring of Graphs
Combinatorics, Probability and Computing, 2014
We study the conflict-free chromatic number χCF of graphs from extremal and probabilistic points of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erdős–Rényi random graph G(n,p) and give the asymptotics for p = ω(1/n). We also show that for p ≥ 1/2 the conflict-free chromatic number differs from the domination number by at most 3.