Polyhedral studies on the convex recoloring problem

A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms.     

Anticoloring and separation of graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. The anticoloring problem seeks, roughly speaking, such colorings with many vertices colored in each color. We deal with the anticoloring problem for planar graphs and, using Lipton and Tarjan’s separation algorithm, provide an algorithm with some bound on the error. We also show that, to solve the anticoloring problem for general graphs, it suffices to solve it for connected graphs.     

Coloring some classes of mixed graphs

We consider the coloring problem for mixed graphs, that is, for graphs containing edges and arcs. A mixed coloring c is a coloring such that for every edge [x_i,x_j], c(x_i)≠c(x_j) and for every arc (x_p,x_q), c(x_p)<c(x_q). We will analyse the complexity status of this problem for some special classes of graphs.     

A Reduction of the Anticoloring Problem to Connected Graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. The anticoloring problem seeks, roughly speaking, for such colorings with many vertices colored in each color. We show that, to solve the anticoloring problem with two colors for general graphs, it suffices to solve it for connected graphs.     

On the Minimum Sum Coloring of P 4-Sparse Graphs

In this paper, we study the minimum sum coloring (MSC) problem on P ₄-sparse graphs. In the MSC problem, we aim to assign natural numbers to vertices of a graph such that adjacent vertices get different numbers, and the sum of the numbers assigned to the vertices is minimum. Based in the concept of maximal sequence associated with an optimal solution of the MSC problem of any graph, we show that there is a large sub-family of P ₄-sparse graphs for which the MSC problem can be solved in polynomial time. Moreover, we give a parameterized algorithm and a 2-approximation algorithm for the MSC problem on general P ₄-sparse graphs.     

Four classes of perfectly orderable graphs

A graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy” coloring heuristic delivers an optimal coloring of F. No polynomial-time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh–Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.     

No-hole 2-distant colorings for Cayley graphs on finitely generated abelian groups

A no-hole 2-distant coloring of a graph Γ is an assignment c of nonnegative integers to the vertices of Γ such that |c(v)-c(w)|?2 for any two adjacent vertices v and w, and the integers used are consecutive. Whenever such a coloring exists, define nsp(Γ) to be the minimum difference (over all c) between the largest and smallest integers used. In this paper we study the no-hole 2-distant coloring problem for Cayley graphs over finitely generated abelian groups. We give sufficient conditions for the existence of no-hole 2-distant colorings of such graphs, and obtain upper bounds for the minimum span nsp(Γ) by using a group-theoretic approach.     

A solitaire game played on 2-colored graphs

We introduce a solitaire game played on a graph. Initially one disk is placed at each vertex, one face green and the other red, oriented with either color facing up. Each move of the game consists of selecting a vertex whose disk shows green, flipping over the disks at neighboring vertices, and deleting the selected vertex. The game is won if all vertices are eliminated. We derive a simple parity-based necessary condition for winnability of a given game instance. By studying graph operations that construct new graphs from old ones, we obtain broad classes of graphs where this condition also suffices, thus characterizing the winnable games on such graphs. Concerning two familiar (but narrow) classes of graphs, we show that for trees a game is winnable if and only if the number of green vertices is odd, and for n-cubes a game is winnable if and only if the number of green vertices is even and not all vertices have the same color. We provide a linear-time algorithm for deciding winnability for games on maximal outerplanar graphs. We reduce the decision problem for winnability of a game on an arbitrary graph G to winnability of games on its blocks, and to winnability on homeomorphic images of G obtained by contracting edges at 2-valent vertices.     

Mutual exclusion scheduling with interval graphs or related classes. Part II

This paper is the second part of a study devoted to the mutual exclusion scheduling problem. Given a simple and undirected graph G and an integer k, the problem is to find a minimum coloring of G such that each color is used at most k times. The cardinality of such a coloring is denoted by χ(G,k). When restricted to interval graphs or related classes like circular-arc graphs and tolerance graphs, the problem has some applications in workforce planning. Unfortunately, the problem is shown to be NP-hard for interval graphs, even if k is a constant greater than or equal to four [H.L. Bodlaender, K. Jansen, Restrictions of graph partition problems. Part I. Theoret. Comput. Sci. 148 (1995) 93-109]. In this paper, the problem is approached from a different point of view by studying a non-trivial and practical sufficient condition for optimality. In particular, the following proposition is demonstrated: if an interval graph G admits a coloring such that each color appears at least k times, then χ(G,k)=⌈n/k⌉. This proposition is extended to several classes of graphs related to interval graphs. Moreover, all our proofs are constructive and provide efficient algorithms to solve the MES problem for these graphs, given a coloring satisfying the condition in input.     

Improved bounds on linear coloring of plane graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we give some upper bounds on linear chromatic number for plane graphs with respect to their girth, that improve some results of Raspaud and Wang (2009).     

Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

Bounds on vertex colorings with restrictions on the union of color classes

A proper coloring of a graph is a labeled partition of its vertices into parts which are independent sets. In this paper, given a positive integer j and a family ? of connected graphs, we consider proper colorings in which we require that the union of any j color classes induces a subgraph which has no copy of any member of ?. This generalizes some well‐known types of proper colorings like acyclic colorings (where j = 2 and ?is the collection of all even cycles) and star colorings (where j = 2 and ?consists of just a path on 4 vertices), etc. For this type of coloring, we obtain an upper bound of O(d^{(k ? 1)/(k ? j)}) on the minimum number of colors sufficient to obtain such a coloring. Here, d refers to the maximum degree of the graph and k is the size of the smallest member of ?. For the case of j = 2, we also obtain lower bounds on the minimum number of colors needed in the worst case. As a corollary, we obtain bounds on the minimum number of colors sufficient to obtain proper colorings in which the union of any j color classes is a graph of bounded treewidth. In particular, using O(d^8/7) colors, one can obtain a proper coloring of the vertices of a graph so that the union of any two color classes has treewidth at most 2. We also show that this bound is tight within a multiplicative factor of O((logd)^1/7). We also consider generalizations where we require simultaneously for several pairs (j_i, ?_i) (i = 1, …, l) that the union of any j_i color classes has no copy of any member of ?_i and obtain upper bounds on the corresponding chromatic numbers. © 2011 Wiley Periodicals, Inc. J Graph Theory 66: 213–234, 2011     

Results on the Grundy chromatic number of graphs

Given a graph G, by a Grundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i<j, every vertex of G colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by Γ(G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining Γ(G)?k, for any fixed integer k and show that it is a polynomial time problem. But in general, Grundy number is an NP-complete problem. We show that it is NP-complete even for the complement of bipartite graphs and describe the Grundy number of these graphs in terms of the minimum edge dominating number of their complements. Next we obtain some additive Nordhaus-Gaddum-type inequalities concerning Γ(G) and Γ(G^c), for a few family of graphs. We introduce well-colored graphs, which are graphs G for which applying every greedy coloring results in a coloring of G with χ(G) colors. Equivalently G is well colored if Γ(G)=χ(G). We prove that the recognition problem of well-colored graphs is a coNP-complete problem.     

Anticoloring of a family of grid graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. We deal with the anticoloring problem with two colors, in the particular cases of graphs which are strong products of two paths or two cycles, and provide an explicit optimal solution.     

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.     

Almost all k-colorable graphs are easy to color

We describe a simple and efficient heuristic algorithm for the graph coloring problem and show that for all k ≥ 1, it finds an optimal coloring for almost all k-colorable graphs. We also show that an algorithm proposed by Brélaz and justified on experimental grounds optimally colors almost all k-colorable graphs. Efficient implementations of both algorithms are given. The first one runs in O(n + m log k) time where n is the number of vertices and m the number of edges. The new implementation of Brélaz's algorithm runs in O(m log n) time. We observe that the popular greedy heuristic works poorly on k-colorable graphs.     

Injective colorings of planar graphs with few colors

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥ 19 and maximum degree Δ are injectively Δ-colorable. We also show that all planar graphs of girth ≥ 10 are injectively (Δ+1)-colorable, that Δ+4 colors are sufficient for planar graphs of girth ≥ 5 if Δ is large enough, and that subcubic planar graphs of girth ≥ 7 are injectively 5-colorable.     

Trapezoid graphs and their coloring

We define trapezoid graphs, an extension of both interval and permutation graphs. We show that this new class properly contains the union of the two former classes, and that trapezoid graphs are equivalent to the incomparability graphs of partially ordered sets having interval order dimension at most two. We provide an optimal coloring algorithm for trapezoid graphs that runs in time O(nk), where n is the number of nodes and k is the chromatic number of the graph. Our coloring algorithm has direct applications to channel routing on integrated circuits.     

Locally identifying coloring in bounded expansion classes of graphs

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper bound for the lid-chromatic number of planar graphs. This answers in a positive way a question of Esperet et al. [L. Esperet, S. Gravier, M. Montassier, P. Ochem, A. Parreau, Locally identifying coloring of graphs, Electron. J. Combin. 19 (2) (2012)].     

On coloring problems with local constraints

We show complexity results for some generalizations of the graph coloring problem on two classes of perfect graphs, namely clique trees and unit interval graphs. We deal with the μ-coloring problem (upper bounds for the color on each vertex), the precoloring extension problem (a subset of vertices colored beforehand), and a problem generalizing both of them, the (γ, μ)-coloring problem (lower and upper bounds for the color on each vertex). We characterize the complexity of all those problems on clique trees of different heights, providing polytime algorithms for the cases that are easy. These results have two interesting corollaries: first, one can observe on clique trees of different heights the increasing complexity of the chain k-coloring, μ-coloring, (γ, μ)-coloring, list-coloring. Second, clique trees of height 2 are the first known example of a class of graphs where μ-coloring is polynomial time solvable and precoloring extension is NP-complete, thus being at the same time the first example where μ-coloring is polynomially solvable and (γ, μ)-coloring is NP-complete. Last, we show that the μ-coloring problem on unit interval graphs is NP-complete. These results answer three questions from [Ann. Oper. Res. 169(1) (2009), 3–16].