On Equitable Colorings of Windmill Graphs

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.

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 ℱ 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 ℱ. This generalizes some well-known types of proper colorings like acyclic colorings (where j = 2 and ℱis the collection of all even cycles) and star colorings (where j = 2 and ℱconsists 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 ℱ. 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(d8/7) 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 (ji, ℱi) (i = 1, …, l) that the union of any ji color classes has no copy of any member of ℱi and obtain upper bounds on the corresponding chromatic numbers. © 2011 Wiley Periodicals, Inc. J Graph Theory 66: 213–234, 2011 © 2011 Wiley Periodicals, Inc.

Locating metric coloring is a variation of metric coloring in graphs that integrates vertex coloring with the uniqueness of metric representations. In this coloring, each vertex in a connected graph is assigned a color such that the distance vectors to each color class are distinct for every pair of different vertices. Let be a coloring function (not necessarily proper). The coloring ccc is called a locating metric coloring if, for any two distinct vertices , their distance vectors , so it is obtained represents the partition of vertices by color classes. Thus, for every vertex, the distance vector are different. Vertices may share the same color, whether adjacent or not, as long as their metric representations are unique. The smallest number of colors required for such a coloring is called the locating metric chromatic number, denoted T This study focuses on analyzing locating metric coloring for three specific graphs: the Cherry Blossom graph , the Sun Flower graph , and the Closed Dutch Windmill graph . These graphs were chosen due to the absence of prior research on their locating metric coloring properties. The research method combines pattern recognition and a deductive-axiomatic approach. The proof process begins by determining lower bounds, followed by the construction of upper bounds through coloring function analysis. The resulting locating metric chromatic numbers for each graph are then established.

On some domination colorings of graphs

A proper vertex coloring of a graph is equitable if the sizes of color classes dier by atmost one. The notion of equitable coloring was introduced by Meyer in 1973. A proper h colorable graph K is said to be equitably h-colorable if the vertex sets of K can be partioned into h independent color classes V 1 ; V 2 ; :::; V h such that the condition jjV i j jV j jj 1 holds for all different pairs of i and j and the least integer h is known as equitable chromatic number of K. In this paper, we nd the equitable coloring of book graph, middle, line and central graphs of book graph.

Let [Formula: see text] be a ring with unity. The cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertices are the set of all nonzero and non-unit elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The reduced cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Jesili et al. [On the genus of reduced cozero-divisor graph of commutative rings, Soft Comput. 27(2) (2023) 657–666] classified all non-local commutative rings whose reduced cozero-divisor graph is of genus one. In this paper, we characterize all classes of finite non-local commutative rings whose reduced cozero-divisor graph is of genus two. Moreover, we classify all the non-local commutative rings whose cozero-divisor graph is of genus two.

A mixed graph is, informally, an object obtained from a simple undirected graph by choosing an orientation for a subset of its edges. A mixed graph is $(m, n)$-coloured if each edge is assigned one of $m \geq 0$ colours, and each arc is assigned one of $n \geq 0$ colours. Oriented graphs are $(0, 1)$-coloured mixed graphs, and 2-edge-coloured graphs are $(2, 0)$-coloured mixed graphs. We show that results of Sopena for vertex colourings of oriented graphs, and of Kostochka, Sopena and Zhu for vertex colourings oriented graphs and 2-edge-coloured graphs, are special cases of results about vertex colourings of $(m, n)$-coloured mixed graphs. Both of these can be regarded as a version of Brooks' Theorem.Comment: 7 pages, no figures

Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless G is a complete graph or a cycle with an odd number of vertices, or k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike.

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

An edge of a properly vertex-colored graph is said to be a good edge if it has end vertices of different color. The chromatic completion graph of a graph G is a graph obtained by adding all possible good edges to G. The chromatic completion number of G is the maximum number of new good edges added to G. An equitable coloring of a graph G is a proper vertex coloring of G such that the difference of cardinalities of any two color classes is at most 1. In this paper, we discuss the chromatic completion graphs and chromatic completion number of certain graph classes, with respect to their equitable coloring.

Tree graph is a connected graph and has no circuits. Tree graphs used in this study include: broom graph, firework graph, banana tree graph, centipede graph, E graph, and double star graph. Graph coloring is the process of giving color to graph elements with the rule that neighboring elements must not have the same color and the number of colors used must be as minimal as possible. b-Coloring of a graph G is a coloring of the vertices of G such that each color class has at least one vertex adjacent to all other color classes. The b-Chromatic number of a graph G is denoted by φ (G), is the largest integer k such that G has a b-coloring with k colors. The limit of b-coloring of graph G with maximum degree ∆(G) is as follows. χ(G)≤φ(G)≤∆(G)+1.χ(G) is the chromatic number of a graph G where χ(G) is the minimum value of the color required for proper coloring of graph G. While ∆(G) is the maximum degree of the vertices in graph G. This study uses an exploratory research type with axiomatic deductive method and pattern detection method. Based on the results of this study, the results of the b-coloring analysis on the tree graph family are known. The results of this study are expected to be used as study material and development of scientific knowledge related to b-coloring of other graphs

A proper coloring of graph G is said to be equitable if the number of element(Vertices) in any two color classes differ by atmost one. In equitable coloring the minimum number of color classes is called the equitable chromatic number. In this paper, we found some theorems on equitable coloring and derived the equitable chromatic number of convex polytope graphs with certain pendant edges added.

Families of disjoint sets coloring (partitioning) technique is trial to unify and generalize all types of coloring such as edge coloring, vertex coloring, and face coloring. The concept of standard of partitioning edges into color classes is essential concept to follow and understand families of disjoint sets coloring (partitioning) technique. In this paper, we introduce some examples of standard of partitioning, and introduce some results of partitioning edges. Some of these results explain importance of standard of partitioning related to subgraphs, and some results related to maximum number of edges belongs to each color class. Also, we introduce results of partitioning edges related to quasi non common edges. Here we list some properties related to standard of partitioning edges into color classes.

We consider the relationship between the graph coloring problem (GCP) and the vector assignment problem (VAP). Given an undirected graph, VAP asks to assign a vector to each vertex so as to maximize the minimum angle between the vectors corresponding to adjacent vertices. We show that any solution to the VAP in the 2-dimensional space, which we call the 2-dimensional VAP (2VAP), gives a feasible coloring, and that such transformation can be computed efficiently. We also show that any optimal solution to 2VAP gives an optimal coloring for GCP. Based on this fact, we propose a heuristic algorithm for GCP, whose search space is the set of solutions for 2VAP. The algorithm is quite simple and can be considered as a variant of the threshold accepting. The experiments show that our algorithm works well for graphs with relatively low degree.

A dominated coloring of a graph G is a proper vertex coloring where each color class is dominated by at least one vertex of G. The dominated chromatic number of G, denoted ?dom (G), is the m inimum number of colors required for a dominated coloring of G. In this paper, we provide new bounds for ?dom (G) and characterize all graphs that achieve some of these bounds. Also we investigate graphs G for which ?dom (G) = ?(G) where ?(G) is the chromatic number of G, in particular we give a characterization o ? cubic graphs G such that ?dom (G) = ?(G).