Graph Coloring Research Articles

Graph Coloring Algorithm Based on Minimal Cost Graph Neural Network

Graph Coloring Algorithm Based on Minimal Cost Graph Neural Network

Local antimagic vertex coloring of a Myceilski of graphs

Local antimagic vertex coloring of a Myceilski of graphs

Weighted Recursive Graph Color Coding for Enhanced Load Identification

Weighted Recursive Graph Color Coding for Enhanced Load Identification

Proof of a conjecture on edge coloring of the Kneser graph K(t,2)

Proof of a conjecture on edge coloring of the Kneser graph K(t,2)

Edukasi Zat Pewarna Pangan Berbahaya Di SMA Negeri 1 Sukadana

Introduction: In general, food coloring can be obtained from natural dyes and safe synthetic food dyes. Natural dyes are sometimes considered less stable and less durable than synthetic food dyes. Although there are synthetic dyes that are safe for food, there are still people who use materials that are not suitable for use in food. The use of harmful food dyes must be monitored and recognized for their existence. Harmful food dyes that are commonly found are textile dyes such as Rhodamine B or methanyl yellow. Because it is not a proper food coloring, consumption of foods containing this ingredient can cause a series of health problems for those who consume it, especially in a relatively long period of time. This education aims to allow students to distinguish foods that use natural coloring agents and artificial coloring agents. The results of this activity students can find out the characteristics of foods that use harmful dyes and know the prevention that can be done.
 Objective: The purpose of this service is for SMAN 1 SUKADANA students to know the dangers of using hazardous coloring agents in food.
 Method: The method used is providing education by providing material to students, tests before and after material on the Use of Harmful Coloring Substances in Food.
 Result: Increased knowledge and attitudes of students are tested through pre-test and post-test. There has been an increase in knowledge in students about the dangers of using harmful dyes in food in general by 20%, so that the realization of the purpose of this service is expected to be able to distinguish food ingredients that contain harmful coloring agents in food and those that do not contain harmful coloring agents in food. Basically, knowledge and attitudes also depend on the high or low level of education of students. Therefore, the existence of this counseling can help students in adding insight into the dangers of using harmful dyes. The implementation of this activity can run well due to positive support in the implementation process, where support from various parties and there are no obstacles whatsoever.
 Conclusion: The socialization activity on the use of dangerous coloring substances in food to students at SMAN 1 Sukadana received a good response from the school and also students. With outreach activities regarding this, students will understand and be more careful in choosing food to avoid these chemicals.

GRACEFUL CHROMATIC NUMBER OF SOME CARTESIAN PRODUCT GRAPHS

A graph \(G(V,E)\) is a system consisting of a finite non empty set of vertices \(V(G)\) and a set of edges \(E(G)\). A (proper) vertex colouring of \(G\) is a function \(f:V(G)\rightarrow \{1,2,\ldots,k\},\) for some positive integer \(k\) such that \(f(u)\neq f(v)\) for every edge \(uv\in E(G)\). Moreover, if \(|f(u)-f(v)|\neq |f(v)-f(w)|\) for every adjacent edges \(uv,vw\in E(G)\), then the function \(f\) is called graceful colouring for \(G\). The minimum number \(k\) such that \(f\) is a graceful colouring for \(G\) is called the graceful chromatic number of \(G\). The purpose of this research is to determine graceful chromatic number of Cartesian product graphs \(C_m \times P_n\) for integers \(m\geq 3\) and \(n\geq 2\), and \(C_m \times C_n\) for integers \(m,n\geq 3\). Here, \(C_m\) and \(P_m\) are cycle and path with \(m\) vertices, respectively. We found some exact values and bounds for graceful chromatic number of these mentioned Cartesian product graphs.

Star Rainbow Coloring in Graphs

Star Rainbow Coloring in Graphs

R-Chromatic Number On r-Dynamic Vertex Coloring of Comb Graph

Let be a graph with vertex set and edge set . An r-dynamic vertex coloring of a graph is a assigning colors to the vertices of such that for every vertex receives at least colors in its neighbors. The minimum color used in r-dynamic vertex coloring of graph is called the r-dynamic chromatic number denoted as . In this research we well determine the coloring pattern and the r-dynamic chromatic number of the comb graph , central graph of comb graph , middle graph of comb graph , line graph of comb graph , sub-division graf of comb graph , and para-line graph of comb graph Let be a graph with vertex set and edge set . An r-dynamic vertex coloring of a graph is a assigning colors to the vertices of such that for every vertex receives at least colors in its neighbors. The minimum color used in r-dynamic vertex coloring of graph is called the r-dynamic chromatic number denoted as . In this research we well determine the coloring pattern and the r-dynamic chromatic number of the comb graph , central graph of comb graph , middle graph of comb graph , line graph of comb graph , sub-division graf of comb graph , and para-line graph of comb graph

Neighbor Distinguishing Colorings of Graphs with the Restriction for Maximum Average Degree

Neighbor distinguishing colorings of graphs represent powerful tools for solving the channel assignment problem in wireless communication networks. They consist of two forms of coloring: neighbor distinguishing edge coloring, and neighbor distinguishing total coloring. The neighbor distinguishing edge (total) coloring of a graph G is an edge (total) coloring with the requirement that each pair of adjacent vertices contains different color sets. The neighbor distinguishing edge (total) chromatic number of G is the smallest integer k in cases where a neighbor distinguishing edge (total) coloring exists through the use of k colors in G. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs. In this paper, we characterize the neighbor distinguishing edge (total) chromatic numbers of graphs with a maximum average degree less than four by means of the discharging method.

Reconfiguration of vertex colouring and forbidden induced subgraphs

The reconfiguration graph of the k-colourings, denoted Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are adjacent in Rk(G) if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of Rk+1(G) for a k-colourable graph G restricted by forbidden induced subgraphs. We show that Rk+1(G) is connected for every k-colourable H-free graph G if and only if H is an induced subgraph of P4 or P3+P1. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if G is a k-colourable (2K2, C4)-free graph, then Rk+1(G) is connected with diameter at most 4n. Furthermore, we show that Rk+1(G) is connected for every k-colourable (P5, C4)-free graph G.

Distributed Graph Coloring Made Easy

In this article, we present a deterministic \(\mathsf {CONGEST}\) algorithm to compute an O ( k Δ)-vertex coloring in O (Δ / k )+log * n rounds, where Δ is the maximum degree of the network graph and k ≥ 1 can be freely chosen. The algorithm is extremely simple: each node locally computes a sequence of colors and then it tries colors from the sequence in batches of size k . Our algorithm subsumes many important results in the history of distributed graph coloring as special cases, including Linial’s color reduction [Linial, FOCS’87], the celebrated locally iterative algorithm from [Barenboim, Elkin, Goldenberg, PODC’18], and various algorithms to compute defective and arbdefective colorings. Our algorithm can smoothly scale between several of these previous results and also simplifies the state-of-the-art (Δ +1)-coloring algorithm. At the cost of losing some of the algorithm’s simplicity we also provide a O ( k Δ)-coloring algorithm in \(O(\sqrt {\Delta /k})+\log ^{*} n\) rounds. We also provide improved deterministic algorithms for ruling sets, and, additionally, we provide a tight characterization for one-round color reduction algorithms.

On graceful coloring of generalized Petersen graphs

The graphs used in this paper are simple, connected and finite graph. A graceful [Formula: see text]-coloring of a graph [Formula: see text] is a proper vertex coloring [Formula: see text], where [Formula: see text] which induces a proper edge coloring [Formula: see text] that defined by [Formula: see text]. Proper vertex coloring [Formula: see text] of a graph [Formula: see text] is a graceful coloring if [Formula: see text] is a graceful m-coloring for [Formula: see text]. The minimum vertex coloring, that can apply such that a graph [Formula: see text] can be colored with graceful coloring, is called a graceful chromatic number and it is denoted by [Formula: see text]. In this paper, we will investigate the graceful chromatic number of generalized Petersen graph [Formula: see text] for [Formula: see text].

Rainbow subgraphs in edge‐colored complete graphs: Answering two questions by Erdős and Tuza

AbstractAn edge‐coloring of a complete graph with a set of colors is called completely balanced if any vertex is incident to the same number of edges of each color from . Erdős and Tuza asked in 1993 whether for any graph on edges and any completely balanced coloring of any sufficiently large complete graph using colors contains a rainbow copy of . This question was restated by Erdős in his list of “Some of my favourite problems on cycles and colourings.” We answer this question in the negative for most cliques by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph .

On the Independent Coloring of Graphs with Applications to the Independence Number of Cartesian Product Graphs

Let G be a graph with V = V G . A nonempty subset S of V is called an independent set of G if no two distinct vertices in S are adjacent. The union of a class { S : S is an independent set of G } and ∅ is denoted by I G . For a graph H , a function f : V ⟶ I H is called an H − independent coloring of G (or simply called an H − coloring) if f x ∩ f y = ∅ for any adjacent vertices x , y ∈ V and f V is a class of disjoint sets. Let α H , G denote the maximum cardinality of the set{ ∑ x ∈ V f x : f is an H − coloring of G }. In this paper, we obtain basic properties of an H − coloring of G and find α H , G of some families of graphs G and H . Furthermore, we apply them to determine the independence number of the Cartesian product of a complete graph K n and a graph G and prove that α K n □ G = α K n , G .

Harmonious colouring of line graph of commuting and non-commuting graph of D_{2n}

Harmonious coloring of graph \(G\) is a proper vertex coloring, where each pair of colors occurs at most on one pair of adjacent vertices. Minimum number of colors required for Harmonious coloring of \(G\) is the harmonious chromatic number, \(\chi_{H}(G)\). Here we determine the Harmonious chromatic number of the line graph of commuting graph and non-commuting graph of the dihedral group, \(D_{2n}\).

One-dependent colorings of the star graph

One-dependent colorings of the star graph

Average distance colouring of graphs

Abstract For a graph G with n vertices, average distance µ(G) is the ratio of sum of the lengths of the shortest paths between all pairs of vertices to the number of edges in a complete graph on n vertices. In this paper, we introduce average distance colouring and find the average distance colouring number of certain classes of graphs.

PENERAPAN ALGORITMA WELCH-POWELL PADA PENJADWALAN MATA PELAJARAN SD

In the context of scheduling class timetables and allocating teachers to specific subjects in schools, a common issue is the occurrence of schedule conflicts. These conflicts often result in situations where a teacher is assigned to teach the same subject at the same time in different classes, or where teachers are scheduled to teach different subjects simultaneously. The Welch Powell algorithm is a graph coloring method that can be applied to scheduling problems. The scheduling process begins by representing subjects and the teachers assigned to them as a graph. Each subject is represented as a node within the graph, while the edges in the graph represent the classes taught by the subject teachers. The graph coloring procedure starts with the selection of the node with the highest degree in the constructed graph. Through the implementation of the Welch Powell algorithm, it has been observed that graph coloring can be effectively used for scheduling class hours and teacher subject assignments, thus eliminating schedule conflicts.

The number of distinguishing colorings of a Cartesian product graph

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold θ(G) of a graph G is the minimum number of colors k required that any arbitrary k-coloring of G is distinguishing. In this paper, we calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of grids.

An Algorithm for Coloring of Picture Fuzzy Graphs Based on Strong and Weak Adjacencies, and Its Application

The idea of strong and weak adjacencies between vertices has been generalized into fuzzy graphs and intuitionistic fuzzy graphs (IFGs), and it is an important part of making decisions. However, one or two membership degrees are not always sufficient for making decisions on real-world problems that need an answer of types “yes, neutral, and no”. Consequently, in previous work, we generalized the concept into picture fuzzy graphs (PFGs) where each element in the PFG has membership, neutral, and non-membership degrees. Moreover, we constructed the notion of the coloring of PFGs based on strong and weak adjacencies between vertices. In this paper, we investigate some properties of the chromatic number of PFGs based on the concept of strong and weak adjacencies between vertices. According to these properties, we construct an algorithm to find the chromatic number of PFGs. The algorithm is useful when we work with large PFGs. Further, we improve the method to implement the PFG’s coloring for determining traffic signal phasing at an intersection. A case study has also been carried to evaluate the method.