Signed graph coloring

Coloring a signed graph by signed colors, one has a chromatic polynomial with the same enumerative and algebraic properties as for ordinary graphs. New phenomena are the interpretability only of odd arguments and the existence of a second chromatic polynomial counting zero-free colorings. The generalization to voltage graphs is outlined.     

Randomized online graph coloring

We study the problem of coloring graphs in an online manner. The only known deterministic online graph coloring algorithm with a sublinear performance function was found by [9.], 319–325). Their algorithm colors graphs of chromatic number χ with no more than (2χn)/log^* n colors, where n is the number of vertices. They point out that the performance can be improved slightly for graphs with bounded chromatic number. For three-chromatic graphs the number of colors used, for example, is O(n log log log n/log log n). We show that randomization helps in coloring graphs online. We present a simple randomized online algorithm to color graphs with expected number of colors O(2^χχ²n^{(χ−2)/(χ−1)}(log n)^1/(χ−1)). For three-colorable graphs the expected number of colors our algorithm uses is . All our algorithms run in polynomial time. It is interesting to note that our algorithm compares well with the best known polynomial time offline algorithms. For instance, the best polynomial time algorithm known for three-colorable graphs, due to [4.] pp. 554–562). We also prove a lower bound of Ω((1/(χ − 1))((log n/(12(χ + 1))) − 1)^χ−1) for the randomized model. No lower bound for the randomized model was previously known. For bounded χ, our result improves even the best known lower bound for the deterministic case: Ω((log n/log log n)^χ−1), due to Noga Alon (personal communication, September 1989).     

Mixed graph edge coloring

We are interested in coloring the edges of a mixed graph, i.e., a graph containing unoriented and oriented edges. This problem is related to a communication problem in job-shop scheduling systems. In this paper we give general bounds on the number of required colors and analyze the complexity status of this problem. In particular, we provide NP-completeness results for the case of outerplanar graphs, as well as for 3-regular bipartite graphs (even when only 3 colors are allowed, or when 5 colors are allowed and the graph is fully oriented). Special cases admitting polynomial-time solutions are also discussed.     

Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9

Let $k$ be a positive integer. An adjacent vertex distinguishing (for short, AVD) total-$k$-coloring of a graph $G$ is a proper total-$k$-coloring of $G$ such that any two adjacent vertices have different color sets, where the color set of a vertex $v$ contains the color of $v$ and the colors of its incident edges. It was conjectured that any graph with maximum degree $\Delta$ has an AVD total-($\Delta +3$)-coloring. The conjecture was confirmed for any graph with maximum degree at most 4 and any planar graph with maximum degree at least 10. In this paper, we verify the conjecture for all planar graphs with maximum degree at least 9. Moreover, we prove that any planar graph with maximum degree at least 10 has an AVD total-($\Delta +2$)-coloring and the bound $\Delta +2$ is sharp.     

Injective coloring of planar graphs

A coloring of a graph G is injective if its restriction to the neighborhood of any vertex is injective. The injective chromatic numberχ_i(G) of a graph G is the least k such that there is an injective k-coloring. In this paper we prove that if G is a planar graph with girth g and maximum degree Δ, then (1) χ_i(G)=Δ if either g≥20 and Δ≥3, or g≥7 and Δ≥71; (2) χ_i(G)≤Δ+1 if g≥11; (3) χ_i(G)≤Δ+2 if g≥8.     

k-fold coloring of planar graphs

A k-fold n-coloring of G is a mapping φ: V (G) → Zk(n) where Zk(n) is the collection of all ksubsets of {1,2,...,n} such that φ(u) ∩φ(v) = φ if uv ∈ E(G).If G has a k-fold n-coloring,i.e.,G is k-fold n-colorable.Let the smallest integer n such that G is k-fold n-colorable be the k-th chromatic number,denoted by χk(G).In this paper,we show that any outerplanar graph is k-fold 2k-colorable or k-fold χk(C*)-colorable,where C* is a shortest odd cycle of G.Moreover,we investigate that every planar graph with odd girth at least 10k-9(k 3) can be k-fold (2k + 1)-colorable.     

σ-polynomials and graph coloring

A new class of graph polynomials is defined. Tight bounds on the coefficients of the polynomials are given, and the exact polynomials for several classes of graphs are derived. The relationship of these polynomials to chromatic polynomials and graph coloring is discussed.     

Asymmetric directed graph coloring games

This note generalizes the (a,b)-coloring game and the (a,b)-marking game which were introduced by Kierstead [H.A. Kierstead, Asymmetric graph coloring games, J. Graph Theory 48 (2005) 169-185] for undirected graphs to directed graphs. We prove that the (a,b)-chromatic and (a,b)-coloring number for the class of orientations of forests is b+2 if b≤a, and infinity otherwise. From these results we deduce upper bounds for the (a,b)-coloring number of oriented outerplanar graphs and of orientations of graphs embeddable in a surface with bounded girth.     

Metaheuristics for robust graph coloring

This paper studies a robust graph coloring problem, which is a variant of the classical graph coloring problem, where penalties are charged for non-adjacent vertices that are assigned the same color. The problem can be formulated as an unconstrained quadratic programming problem, and has many applications in industry. Since the problem is known to be strongly NP-complete, we develop a number of metaheuristics for it, which are based on various encoding schemes, neighborhood structures, and search algorithms. Extensive experiments suggest that our metaheuristics with a partition based encoding scheme and an improvement graph based neighborhood outperform other methods tested in our study.     

A new kind of graph coloring

A proper k-coloring C¹,C²,…,C^k of a graph G is called strong if, for every vertex uV(G), there exists an index i{1,2,…,k} such that u is adjacent to every vertex of Cⁱ. We consider classes of strongly k-colorable graphs and show that the recognition problem of is NP-complete for every k4, but it is polynomial-time solvable for k=3. We give a characterization of in terms of forbidden induced subgraphs. Finally, we solve the problem of uniqueness of a strong 3-coloring.     

On coloring numbers of graph powers

The weak $r$-coloring numbers ${\mathrm{wcol}}_r\left(G\right)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $\mathrm{col}\left(G\right)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs.Let $G^p$ denote the $p$th power of $G$. We show that, all integers $p>0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta \left(G\right)\le \Delta$ satisfy $\mathrm{col}\left(G^p\right)\in O\left(p\cdot {\mathrm{wcol}}_{\lceil p∕2\rceil }\left(G\right){(\Delta -1)}^{\lfloor p∕2\rfloor }\right)$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2<\mathrm{mad}\left(G\right)\le 2k$, then $\mathrm{col}\left(G^2\right)\le (2k-1)\Delta \left(G\right)+2k+1$.     

Variable space search for graph coloring

Let G=(V,E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,…,k}) to each vertex of G so that no edge has both endpoints with the same color. We propose a new local search methodology, called Variable Space Search, which we apply to the k-coloring problem. The main idea is to consider several search spaces, with various neighborhoods and objective functions, and to move from one to another when the search is blocked at a local optimum in a given search space. The k-coloring problem is thus solved by combining different formulations of the problem which are not equivalent, in the sense that some constraints are possibly relaxed in one search space and always satisfied in another. We show that the proposed algorithm improves on every local search used independently (i.e., with a unique search space), and is competitive with the currently best coloring methods, which are complex hybrid evolutionary algorithms.     

Directed defective asymmetric graph coloring games

We consider the following 2-person game which is played with an (initially uncolored) digraph D, a finite color set C, and nonnegative integers a, b, and d. Alternately, player I colors a vertices and player II colors b vertices with colors from C. Whenever a player colors a vertex v, all in-arcs (w,v) that do not come from a vertex w previously colored with the same color as v are deleted. For each color i the defect digraphD_i is the digraph induced by the vertices of color i at a certain state of the game. The main rule the players have to respect is that at every time for any color i the digraph D_i has maximum total degree of at most d. The game ends if no vertex can be colored any more according to this rule. Player I wins if D is completely colored at the end of the game, otherwise player II wins. The smallest cardinality of a color set C with which player I has a winning strategy for the game is called . This parameter generalizes several variants of Bodlaender’s game chromatic number. We determine the tight (resp., nearly tight) upper bound (resp., ) for the d-relaxed (a,b)-game chromatic number of orientations of forests (resp., undirected forests) for any d and a≥b≥1. Furthermore we prove that these numbers cannot be bounded in case a<b.     

A Branch-and-Cut algorithm for graph coloring

In this paper a Branch-and-Cut algorithm, based on a formulation previously introduced by us, is proposed for the Graph Coloring Problem. Since colors are indistinguishable in graph coloring, there may typically exist many different symmetrical colorings associated with a same number of colors. If solutions to an integer programming model of the problem exhibit that property, the Branch-and-Cut method tends to behave poorly even for small size graph coloring instances. Our model avoids, to certain extent, that bottleneck. Computational experience indicates that the results we obtain improve, in most cases, on those given by the well-known exact solution graph coloring algorithm Dsatur.     

3-dynamic coloring of planar triangulations

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that any vertex $v$ has at least $min\{r,{deg}_G\left(v\right)\}$ distinct colors in $N_G\left(v\right)$. The $r$-dynamic chromatic number${\chi }_r^d\left(G\right)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$.Loeb et al. (2018) showed that if $G$ is a planar graph, then ${\chi }_3^d\left(G\right)\le 10$, and there is a planar graph $G$ with ${\chi }_3^d\left(G\right)=7$. Thus, finding an optimal upper bound on ${\chi }_3^d\left(G\right)$ for a planar graph $G$ is a natural interesting problem. In this paper, we show that ${\chi }_3^d\left(G\right)\le 5$ if $G$ is a planar triangulation. The upper bound is sharp.     

A memetic algorithm for graph coloring

Given an undirected graph G=(V,E)$G=(V\text{,}E)$ with a set V of vertices and a set E of edges, the graph coloring problem consists of partitioning all vertices into k independent sets and the number of used colors k is minimized. This paper presents a memetic algorithm (denoted by MACOL) for solving the problem of graph coloring. The proposed MACOL algorithm integrates several distinguished features such as an adaptive multi-parent crossover (AMPaX) operator and a distance-and-quality based replacement criterion for pool updating. The proposed algorithm is evaluated on the DIMACS challenge benchmarks and computational results show that the proposed MACOL algorithm achieves highly competitive results, compared with 11 state-of-the-art algorithms. The influence of some ingredients of MACOL on its performance is also analyzed.     

Improved square coloring of planar graphs

Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac{3}{2}\mathrm{\Delta }+1$ colors are sufficient to square color every planar graph of maximum degree Δ. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2\mathrm{\Delta }+7$ colors are sufficient, which improves the best known bounds when $6?\mathrm{\Delta }?31$.