On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0-1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.     

Total Colorings Of Degenerate Graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤s. We prove that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.     

Complete Multipartite Graphs and the Relaxed Coloring Game

Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted ?? _g ^d (G). It is known that there exist graphs such that ?? _g ⁰(G)?=?3, but ?? _g ¹(G)?>?3. We will show that for all positive integers m, there exists a complete multipartite graph G such that m?????? _g ⁰(G)?<??? _g ¹(G).     

A Characterization of b‐Perfect Graphs

A b‐coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b‐chromatic number of a graph G is the largest integer k such that G admits a b‐coloring with k colors. A graph is b‐perfect if the b‐chromatic number is equal to the chromatic number for every induced subgraph of G. We prove that a graph is b‐perfect if and only if it does not contain as an induced subgraph a member of a certain list of 22 graphs. This entails the existence of a polynomial‐time recognition algorithm and of a polynomial‐time algorithm for coloring exactly the vertices of every b‐perfect graph. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:95–122, 2012     

List total colorings of series-parallel graphs

A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L(v)|min{5,Δ+1} for each vertex v and |L(e)|max{5,d(v)+1,d(w)+1} for each edge e=vw, where Δ is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with Δ+1 colors if Δ4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition.     

A survey of stratified domination in graphs

A graph G is 2-stratified if its vertex set is partitioned into two nonempty classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The F-domination number of a graph G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that for every blue vertex v of G, there is a copy of F in G rooted at v. In this paper, we survey recent results on the F-domination number for various 2-stratified graphs F.     

The relaxed game chromatic number of outerplanar graphs

The (r,d)‐relaxed coloring game is played by two players, Alice and Bob, on a graph G with a set of r colors. The players take turns coloring uncolored vertices with legal colors. A color α is legal for an uncolored vertex u if u is adjacent to at most d vertices that have already been colored with α, and every neighbor of u that has already been colored with α is adjacent to at most d – 1 vertices that have already been colored with α. Alice wins the game if eventually all the vertices are legally colored; otherwise, Bob wins the game when there comes a time when there is no legal move left. We show that if G is outerplanar then Alice can win the (2,8)‐relaxed coloring game on G. It is known that there exists an outerplanar graph G such that Bob can win the (2,4)‐relaxed coloring game on G. © 2004 Wiley Periodicals, Inc. J Graph Theory 46:69–78, 2004     

Relaxed very asymmetric coloring games

This paper investigates a competitive version of the coloring game on a finite graph G. An asymmetric variant of the (r,d)-relaxed coloring game is called the (r,d)-relaxed (a,b)-coloring game. In this game, two players, Alice and Bob, take turns coloring the vertices of a graph G, using colors from a set X, with |X|=r. On each turn Alice colors a vertices and Bob colors b vertices. A color α∈X is legal for an uncolored vertex u if by coloring u with color α, the subgraph induced by all the vertices colored with α has maximum degree at most d. Each player is required to color an uncolored vertex legally on each move. The game ends when there are no remaining uncolored vertices. Alice wins the game if all vertices of the graph are legally colored, Bob wins if at a certain stage there exists an uncolored vertex without a legal color. The d-relaxed (a,b)-game chromatic number, denoted by , of G is the least r for which Alice has a winning strategy in the (r,d)-relaxed (a,b)-coloring game.The (r,d)-relaxed (1,1)-coloring game has been well studied and there are many interesting results. For the (r,d)-relaxed (a,1)-coloring game, this paper proves that if a graph G has an orientation with maximum outdegree k and a≥k, then for all d≥k²+2k; If a≥k³, then (a,1)- for all d≥2k+1.     

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.     

The convex dimension of a graph

The convex dimension of a graph G=(V,E) is the smallest dimension d for which G admits an injective map f:V?R^d of its vertices into d-space, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admits a map as above such that the images of the vertices of G are also in convex position. In this paper we study the convex and strong convex dimensions of graphs.     

6‐Star‐Coloring of Subcubic Graphs

A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two adjacent vertices are assigned the same color) such that no path on four vertices is 2‐colored. The star chromatic number of G is the smallest integer k for which G admits a star coloring with k colors. In this paper, we prove that every subcubic graph is 6‐star‐colorable. Moreover, the upper bound 6 is best possible, based on the example constructed by Fertin, Raspaud, and Reed (J Graph Theory 47(3) (2004), 140–153).     

A note on the weakly convex and convex domination numbers of a torus

The distanced_G(u,v) between two vertices u and v in a connected graph G is the length of the shortest (u,v) path in G. A (u,v) path of length d_G(u,v) is called a (u,v)-geodesic. A set X⊆V is called weakly convex in G if for every two vertices a,b∈X, exists an (a,b)-geodesic, all of whose vertices belong to X. A set X is convex in G if for all a,b∈X all vertices from every (a,b)-geodesic belong to X. The weakly convex domination number of a graph G is the minimum cardinality of a weakly convex dominating set of G, while the convex domination number of a graph G is the minimum cardinality of a convex dominating set of G. In this paper we consider weakly convex and convex domination numbers of tori.     

Game chromatic number of strong product graphs

The graph coloring game is a two-player game in which the two players properly color an uncolored vertex of G alternately. The first player wins the game if all vertices of G are colored, and the second wins otherwise. The game chromatic number of a graph G is the minimum integer k such that the first player has a winning strategy for the graph coloring game on G with k colors. There is a lot of literature on the game chromatic number of graph products, e.g., the Cartesian product and the lexicographic product. In this paper, we investigate the game chromatic number of the strong product of graphs, which is one of major graph products. In particular, we completely determine the game chromatic number of the strong product of a double star and a complete graph. Moreover, we estimate the game chromatic number of some King's graphs, which are the strong products of two paths.     

On b-perfect Chordal Graphs

The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbour in all other color classes. We characterize the class of chordal graphs for which the b-chromatic number is equal to the chromatic number for every induced subgraph. This research was supported by Algerian-French program CMEP/Tassili 05 MDU 639.     

Vertex colorings without isolates

Call a vertex of a vertex-colored simple graph isolated if all its neighbors have colors other than its own. A. J. Goldman has asked: When is it possible to color b vertices of a graph black and the remaining w vertices white so that no vertex is isolated? We prove (1) if G is connected and has minimum degree 2, it is always possible unless b or w is 1; (2) if G is 2-connected, then for any pair (b, w) there is a coloring in which both monochromatic subgraphs are connected; (3) if G has vertices of degree 1, a necessary condition for a (b, w) coloring without isolates to exist is that there be a solution to a certain knapsack inequality. Next, statements generalizing (1) and (2) to n colors are presented, and current knowledge about their truth is discussed. Then various refinements of (1) and (3), more complicated to state and prove, are given. For instance, with the hypotheses of (1) at least one of the monochromatic subgraphs may be chosen to be connected. Also, the necessary knapsack inequality of (3) is, in most cases, sufficient. Throughout, some consideration is given to the algorithmic complexity of coloring (if possible) without isolates. For most graphs which might arise in practice there is an efficient algorithm for the 2-color problem. However, for arbitrary graphs the 2-(or more) color problem is NP-complete.     

The harmonious coloring problem is NP-complete for interval and permutation graphs

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.     

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.     

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).     

On the Convexity Number of Graphs

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number.     

Nordhaus-Gaddum results for the convex domination number of a graph

The distance d _G (u, v) between two vertices u and v in a connected graph G is the length of the shortest uv-path in G. A uv-path of length d _G (u, v) is called a uv-geodesic. A set X is convex in G if vertices from all ab-geodesics belong to X for any two vertices a, b ?? X. The convex domination number ??_con(G) of a graph G equals the minimum cardinality of a convex dominating set. In the paper, Nordhaus-Gaddum-type results for the convex domination number are studied.