Equitable Coloring Research Articles

Equitable List Coloring of Planar Graphs With Given Maximum Degree

ABSTRACTIf is a list assignment of colors to each vertex of an ‐vertex graph , then an equitable ‐coloring of is a proper coloring of vertices of from their lists such that no color is used more than times. A graph is equitably ‐choosable if it has an equitable ‐coloring for every ‐list assignment . In 2003, Kostochka, Pelsmajer, and West (KPW) conjectured that an analog of the famous Hajnal–Szemerédi Theorem on equitable coloring holds for equitable list coloring, namely, that for each positive integer every graph with maximum degree at most is equitably ‐choosable. The main result of this paper is that for each and each planar graph , a stronger statement holds: if the maximum degree of is at most , then is equitably ‐choosable. In fact, we prove the result for a broader class of graphs—the class of the graphs in which each bipartite subgraph with has at most edges. Together with some known results, this implies that the KPW Conjecture holds for all graphs in , in particular, for all planar graphs. We also introduce the new stronger notion of strongly equitable (SE, for short) list coloring and prove all bounds for this parameter. An advantage of this is that if a graph is SE ‐choosable, then it is both equitably ‐choosable and equitably ‐colorable, while neither of being equitably ‐choosable and equitably ‐colorable implies the other.

Equitable Color Class Domination Number of Honeycomb Networks Graph and Inverse Equitable Color Class Domination Number of a Graph

Objectives: In this study, we find a new graph domination parameter named “Inverse Equitable Color Class Domination” and discuss the equitable color class domination number of honeycomb networks graph. Methods: Let be a connected graph. Assume a group containing colors. Let be an equitably colorable function. A dominating subset of is called an equitable color class dominating set if the number of dominating nodes in each color class is equal. The least possible cardinality of an equitable color class dominating set of is the equitable color class domination number itself. It is indicated by . If there exists another equitable color class dominating set in is called an inverse equitable color class dominating set of corresponding to . It is indicated by . Findings: In this paper we conclude the equitable color class domination number of honeycomb networks, honeycomb cup networks graphs and the inverse equitable color class domination number of some graphs. Novelty: This study introduces the concept of “Inverse Equitable Color Class Domination in Graphs”. It obtains many bounds in terms of nodes, links, and other different parameters. 2020 Mathematics Subject Classification: 05C15, 05C69 Keywords: Dominating Set, Equitable Coloring, Color Class (CC), Equitable Color Class (ECC), Equitable Color Class Dominating Set, Inverse Equitable Color Class Domination Number

List Equitable Coloring of Planar Graphs without $4$- and $6$-Cycles when $\Delta(G)=5$

A graph $G$ is $k$ list equitably colorable, if for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. In 2009, Li and Bu obtained that for planar graph $G$, if $\Delta(G)\geq6$ and without $4$- and $6$-cycles, then $G$ is $\Delta(G)$ list equitably colorable. In order to further prove the conjecture of list equitable coloring, in this paper, we focus on planar graph with $\Delta(G)=5$, and prove that if $G$ is a planar graph without $4$- and $6$-cycles, then $G$ is $\Delta(G)$ list equitably colorable.

Equitable colorings of ��-corona products of cubic graphs

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by x=( G). In this paper the problem of determining the value of equitable chromatic number for multicoronas of cubic graphs G◦ lH is studied. The problem of ordinary coloring of multicoronas of cubic graphs is solvable in polynomial time. The complexity of equitable coloring problem is an open question for these graphs. We provide some polynomially solvable cases of cubical multicoronas and give simple linear time algorithms for equitable coloring of such graphs which use at most x=( G◦ lH) + 1 colors in the remaining cases.

New Results on the Robust Coloring Problem

Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to capture natural constraints of those optimization problems by combining the information provided by two colorings: a vertex coloring of a graph and the induced edge coloring on a subgraph of its complement; the goal is to minimize, among all proper colorings of the graph for a fixed number of colors, the number of edges in the subgraph with the endpoints of the same color. The study of the robust coloring model has been focused on the search for heuristics due to its NP-hard character when using at least three colors, but little progress has been made in other directions. We present a new approach on the problem obtaining the first collection of non-heuristic results for general graphs; among them, we prove that robust coloring is the model that better approaches the equitable partition of the vertex set, even when the graph does not admit a so-called equitable coloring. We also show the NP-completeness of its decision problem for the unsolved case of two colors, obtain bounds on the associated robust coloring parameter, and solve a conjecture on paths that illustrates the complexity of studying this coloring model.

Parameterized complexity for iterated type partitions and modular-width

This paper deals with the complexity of some natural graph problems parameterized by some measures that are restrictions of clique-width, such as modular-width and neighborhood diversity. We introduce a novel parameter, called iterated type partition number, that can be computed in linear time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parameterized by the iterated type partition number. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parameterized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity.Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition number, which enables us to find optimal solutions for several graph problems. As an example, in this paper, we present algorithms parameterized by the iterated type partition number of the input graph for some generalized versions of the Maximum Clique, Minimum Graph Coloring, (Total) Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems. Each algorithm outputs not only the optimal value but also the optimal solution. We stress that while the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. We finally show that the proposed scheme can be used to devise polynomial kernels, with respect to iterated type partition number, for the decisional version of most of the problems mentioned above.

Equitable coloring of planar graphs with maximum degree at least eight

The Chen-Lih-Wu Conjecture states that each connected graph with maximum degree Δ≥3 that is not the complete graph KΔ+1 or the complete bipartite graph KΔ,Δ admits an equitable coloring with Δ colors. For planar graphs, the conjecture has been confirmed for Δ≥13 by Yap and Zhang and for 9≤Δ≤12 by Nakprasit. In this paper, we present a proof that confirms the conjecture for graphs embeddable into a surface with non-negative Euler characteristic with maximum degree Δ≥9 and for planar graphs with maximum degree Δ≥8.

Equitable coloring of planar graphs without 5-cycles and chordal 4-cycles

An equitable [Formula: see text]-coloring of a graph [Formula: see text] is a proper vertex coloring such that [Formula: see text] for any [Formula: see text], where [Formula: see text] ([Formula: see text]) is the set of vertices colored with [Formula: see text]. If there is an equitable [Formula: see text]-coloring of [Formula: see text], then the graph [Formula: see text] is said to be equitably [Formula: see text]-colorable. In this paper, we prove that every planar graph without 5-cycles and chordal 4-cycles has an equitable [Formula: see text]-coloring for [Formula: see text] where [Formula: see text] is the maximum degree of [Formula: see text].

Gap one bounds for the equitable chromatic number of block graphs

An equitable coloring of a graph G is a proper vertex coloring of G such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Andersen–Goldberg–Seymour conjecture for multigraphs. The results obtained in the paper support our conjecture. More precisely, we verify it in the class of block graphs in which each vertex belongs to a maximum independent set. We also show that the conjecture is true for block graphs which contain a vertex that does not lie in an independent set of size larger than two. Finally, we verify the conjecture for some symmetric-like block graphs. In order to derive our results we obtain structural characterizations of block graphs from these classes.

Equitable Coloring of IC-Planar Graphs with Girth g ≥ 7

An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be equitably k-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth g≥7 is equitably Δ(G)-colorable, where Δ(G) is the maximum degree of G.

BILANGAN KROMATIK EQUITABLE PADA GRAF BINTANG, GRAF LOLIPOP, DAN GRAF PERSAHABATAN

Let G be a connected and undirected graph. Vertex coloring in a graph G is a mapping from the set of vertices in G to the set of colors such that every two adjacent vertices have different colors. There are many types of vertex coloring, such as complete coloring, k-differential coloring, and equitable coloring. Equitable coloring of G is a vertex coloring of G that satisfies the condition that for each induced color class it has an equitable cardinality with difference 0 or 1. The minimum number of colors used for such coloring of G is called the equitable chromatic number of G, denoted by χe(G). In this study, we only concern with graphs that have a central vertex, which means a vertex that is adjacent to every other vertex, in particular on the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). This research aims to formulate the equitable chromatic number of the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). The first step taken in this research is to apply vertex coloring to Sn, Ln, and fn. After that, the color classes of the vertex set are obtained and its cardinality is determined. Next, analyze that the applied vertex coloring meets the definition of equitable coloring. Then, prove that the number of colors used is minimum. Thus, the chromatic number for each graph is obtained and proved. Based on this research, the equitable chromatic number of Sn is ⌈n/2⌉ + 1, the equitable chromatic number of Ln is n, and the equitable chromatic number of fn is 3, for n = 1 and n + 1, for n ≥ 2.

Equitable oriented coloring

AbstractIn this paper, we consider equitable oriented colorings of graphs. Such coloring is a natural combination of two well‐known colorings: oriented coloring and equitable coloring. An oriented ‐coloring of an oriented graph is an arc‐preserving homomorphism from into an oriented graph on vertices, or colors. We say that the coloring is equitable if for every two colors , we have . This paper presents the following results: we show that there is an undirected graph such that every orientation of is equitably oriented 5‐colorable, and we show that there is an orientation of that cannot be colored with six colors. In other words, there is a gap in the set of numbers of colors that can be used for equitable oriented coloring of . Additionally, we concentrate on paths and show that there is an orientation of a path that requires four colors to be equitably oriented colorable. On the other hand, we show that every orientation of any path is equitably oriented ‐colorable, for every .

Equitable colouring and scheduling on identical machines

Equitable colouring and scheduling on identical machines

Structural Parameterizations for Equitable Coloring: Complexity, FPT Algorithms, and Kernelization

Structural Parameterizations for Equitable Coloring: Complexity, FPT Algorithms, and Kernelization

Some new results on colour-induced signed graphs

AbstractA signed graph is a graph in which positive or negative signs are assigned to its edges. We consider equitable colouring and Hamiltonian colouring to obtain induced signed graphs. An equitable colour-induced signed graph is a signed graph constructed from a given graph in which each edge uv receives a sign (−1)|c(v)−c(u)|,where c is an equitable colouring of vertex v. A Hamiltonian colour-induced signed graph is a signed graph obtained from a graph G in which for each edge e = uv, the signature function σ(uv)=(−1)|c(v)−c(u)|, gives a sign such that, |c(u)− c(v)| ≥ n − 1 − D(u, v) where c is a function that assigns a colour to each vertex satisfying the given condition. This paper discusses the properties and characteristics of signed graphs induced by the equitable and Hamiltonian colouring of graphs.

Graph theoretic and algorithmic aspect of the equitable coloring problem in block graphs

An equitable coloring of a graph $G=(V,E)$ is a (proper) vertex-coloring of $G$, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.

Structural parameterizations of budgeted graph coloring

We introduce a variant of the graph coloring problem, which we denote as Budgeted Coloring Problem (BCP). Given a graph G, an integer c and an ordered list of integers (b1,b2,…,bc), BCP asks whether there exists a proper coloring of G where the i-th color is used to color at most bi many vertices. This problem generalizes two well-studied graph coloring problems, Bounded Coloring Problem (BoCP) and Equitable Coloring Problem (ECP) and as in the case of other coloring problems, the BCP is NP-hard even for constant values of c. So we study BCP under the paradigm of parameterized complexity, particularly with respect to (structural) parameters that specify how far (the deletion distance) the input graph is from a tractable graph class.•We show that BCP is FPT (fixed-parameter tractable) parameterized by the vertex cover size. This generalizes a similar result for ECP and immediately extends to the BoCP, which was earlier not known.•We show that BCP is polynomial time solvable for cluster graphs generalizing a similar result for ECP. However, we show that BCP is FPT, but unlikely to have polynomial kernel, when parameterized by the deletion distance to clique, contrasting the linear kernel for ECP for the same parameter.•While the BoCP is known to be polynomial time solvable on split graphs, we show that BCP is NP-hard on split graphs. As BoCP is hard on bipartite graphs when c>3, the result follows for BCP as well. We provide a dichotomy result by showing that BCP is polynomial time solvable on bipartite graphs when c=2. We also show that BCP is NP-hard on co-cluster graphs, contrasting the polynomial time algorithm for ECP and BoCP. Finally we present an O⁎(2|V(G)|) algorithm for the BCP, generalizing the known algorithm with a similar bound for the standard chromatic number.

Equitable chromatic number of weak modular product of Some graphs

An equitable coloring of a graph G is a proper coloring of the vertices of G such that the number of vertices in any two color clases differ by at most one. The equitable chromatic number χ=(G) of a graph G is the minimum number of colors needed for an equitable coloring of G. In this paper, we obtain the equitable chromatic number of weak modular product of two graphs G and H, denoted by G o H. First, we consider the graph G o H, where G is the path graph, and H be any simple graph like the path, the cycle graph, the complete graph. Secondly, we consider G and H as the complete graph and cycle graph respectively. Finally, we consider G as the star graph and H be the complete graph and star graph.

Equitable Colorings of Hypergraphs with r Colors

The paper deals with a problem concerning equitable vertex colorings of uniform hypergraphs, i.e., colorings under which there are no monochromatic edges and simultaneously all the color classes have almost the same cardinalities. We obtain a new bound on the edge number of an n-uniform hypergraph that guarantees the existence of an equitable vertex coloring with r colors for this hypergraph.

Improving lower bounds for equitable chromatic number

In many practical applications the underlying graph must be as equitable colored as possible. A coloring is called equitable if the sizes of any two color classes differ by at most one, and the least number of colors for which a graph has such a coloring is called its equitable chromatic number.We introduce a new integer linear programming approach for studying the equitable coloring number of a graph and show how to use it for improving lower bounds for this number. The two stage method is based on finding or upper bounding the maximum cardinality of an equitable color class in a valid equitable coloring and, then, sequentially improving the lower bound for the equitable coloring number.The computational experiments were carried out on DIMACS graphs and other graphs from the literature. The lower bounds were improved for more than half of the instances considered.