Determining the irregular chromatic number of graphs using a rank-based genetic algorithm

This article discusses irregular coloring. Irregular coloring was first introduced by Mary Radcliffe and Ping Zhang in 2007. The coloring c is called irregular coloring if distinct vertices of G have distinct codes. The color code of a vertex v of G with respect to c is code(v) = (a0, a1, a2,…, a k) = a0a1a2, …ak, where a0 = c(v) and ai, (1 < i < k) is the number of vertices that are adjacent to v and colored i. The minimum k-color used in irregular coloring is called the irregular chromatic number and is denoted by \\ir. Irregular coloring is included in proper coloring, where each vertex that is the neighbors must not be the same color. The graphs used in this article are a family of bipartite graphs and a family of tree graphs, including complete bipartite graphs, crown graphs, star graphs, centipede graphs, and double star graphs.

Irregular coloring is a proper coloring and each vertex on a graph must have a different code. The color code of a vertex v is where and is the number of vertices that are adjacent to v and colored i. The minimum k-color used in irregular coloring is called the irregular chromatic number and denoted by . In this paper, we discuss the irregular chromatic number for the bull graph, pan graph, sun graph, peach graph, and caveman graph.

This paper aims at analyzing vertices of proper coloring and several kinds of vertex colorings. The coloring vertices include chief types such as recognizable colorings [1], detectable colorings [2] and, irregular colorings [3]. These colorings are distinctive in some way or other. Regarding irregular colorings, there is a slight deviation which is deemed as Neighborhood distinguishing colorings. This particular coloring has the distinguishing element in that the vertex does not relate to the color assigned to it. The irregular coloring is commonly identified in any sort of graph. This is not the case with the Neighborhood Distinguishing Coloring, since it can be identified in a graph under the condition that if and only if any two non-adjacent vertices do not have the same neighbor. The basic definition and the properties related to this Neighborhood Distinguishing Coloring are illustrated in [5]. Further, this paper attempts to determine the NDC-number of sum of the graphs, Corona of a graph and neighborhood irregular graphs.

Non-dominated sorting genetic algorithm II (NSGA-II) obtains a great success for solving multi-objective optimization problems (MOPs). It uses a tournament selection operator (TSO) to select the suitable individuals for the next generation. However, TSO selects individuals based on the non-dominated rank and the crowding distance of each individual, which exhausts a lot of computational burden. In order to relieve the heavy computational burden, this paper proposes an improved selection operator (ISO) that is based on two selection schemes, i.e., a rank-based selection (S-Rank) and a random-based selection (S-Rand). S-Rank is a scheme that selects individuals based on its non-dominated ranks, in which if the individuals have the different non-dominated ranks, the individuals with lower (better) ranks will be selected for the next generation. On the contrary, if the individuals have the same rank, we first select an objective randomly from all objectives, and then select the individual with the better fitness on this objective to enter the next generation. This is the S-Rand scheme that can increase the diversity of individuals (solutions) due to the random selection of objective. The proposed ISO only calculates the crowding distance of the last (selected) rank individual, and avoids the calculation of the crowding distance of all individuals. The performance of ISO is tested on two different benchmark sets: the ZDT test set and the UF test set. Experimental results show that ISO effectively reduces the computational burden and enhance the selection diversity by the aid of S-Rank and S-Rand.

An irregular coloring is a proper coloring in which distinct vertices have different color codes. In this paper, we obtain the irregular chromatic number for middle graph, total graph, central graph and line graph of wheel graph, helm graph, gear graph and closed helm graph.

Exact square coloring of subcubic planar graphs

Let G = (V, E) be a nontrivial graph and k be a positive integer. Let c : V(G) → {1,2,3,…,k} be a vertex coloring of G such that if uv ∈ E(G) then c(u) ≠ c(v). For 1 ≤ i ≤ k, let Si be the ith set of vertices given color i and define Π = {S1 ,S2 ,…,Sk }. The color code of a vertex v ∈ V (G), denoted by cπ(v), is defined as the ordered k-tuple cπ(v) = (d(v, S1 ), d(v, S2), … , d(v, S k)), where d(v, Si) = min{d(v, x) | x ∈ Si} for 1 ≤ i ≤ k. If every two vertices u and v in G have different color codes, then c is defined as the locating k-coloring of G. The minimum number of color used in the locating k-coloring of G is defined as the locating chromatic number of G, denoted by XL(G). This paper determined the locating chromatic number of some Buckminsterfullerene-type graph and some (4, 6)-fullerene graphs.

A proper coloring c of a graph G is an assignment of colors to the vertices of G in such a way that no two adjacent vertices will get the same color. The coloring c is irregular if every vertex in G has different color codes. The minimum number of colors needed to obtain an irregular coloring is the irregular chromatic number. In this paper we study the irregular chromatic number of Join of graphs and Platonic graphs.

Let G be a simple connected graph. A coloring of G is said to be irregular if the vertices receives distinct color codes. The minimum number of color codes required to color the graph G is called the irregular chromatic number and it is denoted by χir (G). In this paper, we obtain the irregular coloring number for certain classes of corona product of graphs and Sierinski graphs.

A proper coloring which means, adjacent vertices have different colors. Irregular coloring was introduced by Radcliffe and Zhang. In this paper, we find the irregular chromatic number for the middle graph, total graph, central graph and line graph of fan graph and the central graph of wheel graph and sunlet graph.

Fullerenes are polyhedral molecules composed solely of carbon atoms, available in various sizes and shapes. These structures can also be depicted as graphs, with the vertices symbolizing the atoms and the edges representing the bonds between them. A fullerene graph is defined as a 3-connected, 3-regular planar graph that consists only of pentagonal and hexagonal faces. This paper examines the perfect 2- and 3-coloring of fullerene graphs, with a particular focus on irreducible fullerenes. The proposed approach begins by obtaining the adjacency matrix of the graphs and then comparing its eigenvalues with those of the parameter matrices. If the eigenvalues of a parameter matrix are a subset of the graph's eigenvalues, we retain these matrices for further analysis to determine their suitability for perfect coloring.

It was seen in the preceding chapter that every connected graph of order 3 or more has an irregular weighting. Therefore, if G is a connected graph of order 3 or more, there exists a weighting w : E(G) → [k] for some integer k ≥ 2 such that the vertices in the resulting weighted graph H of G have distinct degrees. To obtain an irregular weighted graph H with a given underlying graph G, it may very well be necessary for the degrees of the vertices of H to be large, possibly some much larger than the order of G. With this observation in mind, a different weighting of a graph was introduced, referred to as an edge coloring, which was employed to produce an irregular vertex coloring, with the goal of minimizing the largest vertex color. Since irregular colorings are often called rainbow colorings, this is the terminology we use.

Intelligent analysis and designing of network routing provides an edge in this increasingly fast era. In this work, we present a variation of Genetic Algorithm (GA) for finding the Optimized shortest path of the network. The algorithm finds the optimal path by using an objective function consisting of the bandwidth and delay metrics of the network. We also introduce the concept of ¿2-point over 1-point crossover¿. The population comprises of all chromosomes (feasible and infeasible). Moreover, it is of variable length, so that the algorithm can perform efficiently in all scenarios. Rank-based selection is used for cross-over operation. Mutation operation is used for maintaining the population diversity. We have also performed various experiments for the population selection. The experiments indicate that random selection method is the most optimum. Hence, the population is selected randomly once the generation is developed. The results prove our assertion that our proposed algorithm finds the optimal shortest path more efficiently than existing algorithms. In this work, we have shown the results using a smaller network; however the work for larger network is in progress.

In this paper, we present a variation of Genetic Algorithm (GA) for finding the Optimized shortest path of the network. The algorithm finds the optimal path based on the bandwidth and utilization of the network. The main distinguishing element of this work is the use of “2-point over 1-point crossover”. The population comprises of all chromosomes (feasible and infeasible). Moreover, it is of variable length, so that the algorithm can perform efficiently in all scenarios. Rankbased selection is used for cross-over operation. Therefore, the best chromosomes crossover and give the most suitable offsprings. If the resulting offsprings are least fitted, they are discarded. Mutation operation is used for maintaining the population diversity. We have also performed various experiments for the population selection. The experiments indicate that random selection method is the most optimum. Hence, the population is selected randomly once the generation is developed. In this paper, we have shown the results using a smaller network; however the work for larger network is in progress.

The attractive properties of the hypercube graph such as its diameter, good connectivity, and symmetry have made it a popular topology for the design of multi-computer interconnection networks. Efforts to improve some of these properties have led to the evolution of hypercube variants. Let c be the proper coloring of graph G, where the neighboring vertices will get individual colors. Coloring c is irregular if distinct vertices have distinct color codes and the least number of colors that ought to receive an irregular coloring is the irregular chromatic number, χir (G). In this paper, we discuss the irregular coloring and find the irregular chromatic number for the hypercube graph Qn and some of its variants using binomial coefficients for the Locally twisted cube graph LTQn, Crossed cube graph CQn and two types of Fractal cubic network graph FCNG1 (k) and FCNG2 (k).