Graph Coloring Research Articles

Chromatic coloring of distance graphs, III

A graph G(Z, D) with vertex set Z is called an integer distance graph if its edge set is obtained by joining two elements of Z by an edge whenever their absolute difference is a member of D. When D = P or D ⊆ P where P is the set of all prime numbers then we call it a prime distance graph. After establishing the chromatic number of G(Z, P ) as four, Eggleton has classified the collection of graphs as belonging to class i if the chromatic number of G(Z, D) is i. The problem of characterizing the family of graphs belonging to class i when D is of any given size is open for the past few decades. As coloring a prime distance graph is equivalent to producing a prime distance labeling for vertices of G, we have succeeded in giving a prime distance labeling for certain class of all graphs considered here. We have proved that if D = {2, 3, 5, 7, 7th prime, 10th prime, 13th prime, 16th prime, (7 + j)th prime, ..., (4 + j)th prime for any s ∈ N}, then there exists a prime distance graph with distance set D in class 4 and if D = {2, 3, 5, 4th prime, 6th prime, 8th prime, (4 + j)th prime, ..., (2 + j)th prime for any s ∈ N} then there exists a prime distance graphs with distance set D in class 3. Further, we have also obtained some more interesting results that are either general or existential such as a) If D is a specific sequence of integers in arithmetic progression then there exist a prime distance graph with distance set D, b) If G is any prime distance graph in class i for 1 ≤ i ≤ 4 then G × K2 is also a prime distance graph in the respective class i, c) A countable union of disjoint copies of prime distance graph is again a prime distance graph, d) The Middle/Total graph of a path on n vertices is a prime distance graph. In addition we also provide a new different proof for establishing a fact that all cycles are prime distance graph.

On join product and local antimagic chromatic number of regular graphs

Let $$G = (V,E)$$ be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection $$f \colon E \to \{1,2,\ldots,q\}$$ is called a local antimagic labeling of G if for any two adjacent vertices $$u$$ and $$v$$ , we have $$f^+(u) \ne f^+(v)$$ , where $$f^+(u) = \sum_{e\in E(u)} f(e)$$ , and $$E(u)$$ is the set of edges incident to $$u$$ . Thus, any local antimagic labeling induces a proper vertex coloring of G if vertex $$v$$ is assigned the color $$f^+(v)$$ . The local antimagic chromatic number, denoted $$\chi_{la}(G)$$ , is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The join graph of G and H, denoted $$G \vee H$$ , is the graph with $$V(G\vee H) = V(G) \cup V(H)$$ and $$E(G\vee H) = E(G) \cup E(H) \cup \{uv \mid u\in V(G)$$ , $$v \in V(H)\}$$ . In this paper, we investigated $$\chi_{la}(G\vee H)$$ . Consequently, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.

A Makespan-Optimal Schedule for Processing Jobs with Possible Operation Preemptions As an Optimal Mixed Graph Coloring

A Makespan-Optimal Schedule for Processing Jobs with Possible Operation Preemptions As an Optimal Mixed Graph Coloring

Graph Automata and Graph Colorability

Automata recognizing graphs can be constructed by employing the algebraic structure of graphoids. For the construction of a graph automaton, the relations over the Kleene star of the state set must constitute a graphoid. Hence different kinds of graphoids produce graph automata with diverse operation and recognition capacity. In this paper we show that graph colorability is recognized by automata operating over the simplest possible abelian graphoid.

Coloring Bipartite Graphs with Semi-small List Size

Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex’s list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting asymptotics, revealing an invariant quantity instrumental in determining choosability across most of the parameter space. By connecting this quantity to a simple question on independent sets of hypergraphs, we strengthen bounds when a part has list size 2. Finally, we state via our framework a conjecture on general bipartite graphs, unifying three conjectures of Alon–Cambie–Kang.

Improved Approximation Algorithms for Bin Packing with Conflicts

Given a set of items, and a conflict graph defined on the item set, the problem of bin packing with conflicts asks for a partition of items into a minimum number of independent sets so that the total size of items in each independent set does not exceed the bin capacity. As a generalization of both classic bin packing and classic vertex coloring, it is hard to approximate the problem on general graphs. We present new approximation algorithms for bipartite graphs and split graphs. The absolute approximation ratios are shown to be [Formula: see text] and [Formula: see text] respectively, both improving the existing results.

Total Global Dominator Coloring of Trees and Unicyclic Graphs

A total global dominator coloring of a graph is a proper vertex coloring of with respect to which every vertex in dominates a color class, not containing and does not dominate another color class. The minimum number of colors required in such a coloring of is called the total global dominator chromatic number, denoted by . In this paper, the total global dominator chromatic number of trees and unicyclic graphs are explored.

Acyclic Coloring of Certain Graphs

A graph G with acyclic coloring has no two adjacent vertices with the same color and no bichromatic cycle. Also, the coloring results in a forest when any two-color classes are combined. The concept of acyclic coloring plays a pivotal role in the computation of Hessians, Kekule structures classification, coding theory, and statistical mechanics. In this paper, the acyclic chromatic number of generalized fan graph, generalized Möbius ladder graph and flower snark graph have been determined.

Teaching and Learning of Vertex Coloring Using an Inquiry-Based Approach

Traditional lecturing is the dominant method of teaching mathematics in undergraduate mathematical courses in many countries, whereas active student-centered approaches such as inquiry-based learning have been shown in some instances to be more effective. Most teaching resources (inquiry-based tasks) available at the tertiary level are related to calculus, linear algebra, and differential equations courses. In this paper, we focus on using an inquiry-based approach to explore the teaching and learning of vertex coloring, a common topic in graph theory and undergraduate discrete mathematical courses. We designed seven inquiry-based tasks related to vertex coloring to teach vertex coloring in an undergraduate graph theory course to students majoring in mathematics. In sharing these tasks, describing students’ engagement with them, and discussing how these tasks were perceived by the students and lecturer, we hope to promote using inquiry-based tasks in undergraduate discrete mathematical courses.

Robust Coloring Optimization Model on Electricity Circuit Problems

The Graph Coloring Problem (GCP) is assigning different colors to certain elements in a graph based on certain constraints and using a minimum number of colors. GCP can be drawn into optimization problems, namely the problem of minimizing the color used together with the uncertainty in using the color used, so it can be assumed that there is an uncertainty in the number of colored vertices. One of the mathematical optimization techniques in dealing with uncertainty is Robust Optimization (RO) combined with computational tools. This article describes a robust GCP using the Polyhedral Uncertainty Theorem and model validation for electrical circuit problems. The form of an electrical circuit color chart consists of corners (components) and edges (wires or conductors). The results obtained are up to 3 colors for the optimization model for graph coloring problems and up to 5 colors for robust optimization models for graph coloring problems. The results obtained with robust optimization show more colors because the results contain uncertainty. When RO GCP is applied to an electrical circuit, the model is used to place the electrical components in the correct path so that the electrical components do not collide with each other.

Superfast coloring in CONGEST via efficient color sampling

We present a procedure for efficiently sampling colors in the CONGEST model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to Θ(log⁡n) semi-random colors unused by their neighbors in O(1) rounds, even in the distance-2 setting. This yields algorithms with O(log⁎⁡Δ) complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an O(log⁎⁡Δ)-round CONGEST algorithm for (1+ϵ)Δ-edge coloring when Δ=Ω(log1+1/log⁎⁡n⁡n), and a poly(log⁡log⁡n)-round algorithm for (2Δ−1)-edge coloring in general. The sampling procedure is inspired by a seminal result of Newman in communication complexity.

Odd coloring of sparse graphs and planar graphs

An odd c-coloring of a graph is a proper c-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph G with mad(G)≤4c−4c+1 has an odd c-coloring for c≥4, and proved the conjecture for c∈{5,6}. In particular, planar graphs with girth at least 7 and 6 have an odd 5-coloring and an odd 6-coloring, respectively.We completely resolve Cranston's conjecture. For c≥7, we show that the conjecture is true, in a stronger form that was implicitly suggested by Cranston, but for c=4, we construct counterexamples, which all contain 5-cycles. On the other hand, we show that a graph G with mad(G)<229 and no induced 5-cycles has an odd 4-coloring. This implies that a planar graph with girth at least 11 has an odd 4-coloring. We also prove that a planar graph with girth at least 5 has an odd 6-coloring.

Lower bounds on the Erdős–Gyárfás problem via color energy graphs

AbstractGiven positive integers and , a ‐coloring of the complete graph is an edge‐coloring in which every ‐clique receives at least colors. Erdős and Shelah posed the question of determining , the minimum number of colors needed for a ‐coloring of . In this paper, we expand on the color energy technique introduced by Pohoata and Sheffer to prove new lower bounds on this function, making explicit the connection between bounds on extremal numbers and . Using results on the extremal numbers of subdivided complete graphs, theta graphs, and subdivided complete bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer, giving the first nontrivial lower bounds on for some pairs and improving previous lower bounds for other pairs.

3-facial edge-coloring of plane graphs

An ℓ-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most ℓ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the ℓ-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It is conjectured that at most 3ℓ+1 colors suffice for an ℓ-facial edge-coloring of any plane graph. The conjecture has only been confirmed for ℓ≤2, and in this paper, we prove its validity for ℓ=3.

The (3,3)-colorability of planar graphs without 4-cycles and 5-cycles

A graph G is called (d1,…,dr)-colorable if its vertex set can be partitioned into r sets V1,…,Vr such that the maximum degree of the induced subgraph G[Vi] of G is at most di for i∈{1,…,r}. Steinberg conjectured that every planar graph without 4/5-cycles is (0, 0, 0)-colorable. Unfortunately, the conjecture does not hold and it has been proved that every planar graph without 4/5-cycles is (1, 1, 0)-colorable. When only two colors are allowed to use, it is known that some planar graphs without 4/5-cycles are not (1,k)-colorable for any k≥0 and every planar graph without 4/5-cycles is (3,4)-colorable or (2,6)-colorable. In this paper, we reduce the gap for 2-coloring by proving that every planar graph without 4/5-cycles is (3,3)-colorable.

Total dominator chromatic number of Kneser graphs

Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures. A total dominator coloring (briefly, a TDC) of a graph G is a proper coloring (i.e. a partition of the vertex set V(G) into independent subsets named color classes) in which each vertex of the graph is adjacent to all of vertices of some color class. The total dominator chromatic number of G is the minimum number of color classes in a TDC of G. In this paper among some other results and by using the existence of Steiner triple systems, we determine the total dominator chromatic number of the Kneser graph for each

Bipolar Triangular Neutrosophic Chromatic Numbers with the Application of traffic light system

For addressing issues in several domains, such as theoretical computer science, engineering, physics, combinatorics, and the medical sciences, graph theory is a crucial component of mathematics. Graph coloring is one of the new settings that is emerging in a neutrosophic chromatic number environment. In addition to introducing the idea of bipolar triangular neutrosophic chromatic graphs (BTNCG), this work also examines and demonstrates the algebraic assumption. The proposed concept has been applied in a traffic signal system to discover a new lane to avoid traffic in peak hours.

Applying Graph Neural Networks to the Decision Version of Graph Combinatorial Optimization Problems

In recent years, there has been a significant increase in the application of graph neural networks on a wide range of different problems. A specially promising direction of research is on graph convolutional neural networks (GCN). This work focuses on the application of GCNs to graph combinatorial optimization problems (GCOPs), specifically on their decision versions. GCNs are applied on the family of GCOPs in which the objective function is directly related to the number of vertices in a graph. The selected problems are the minimum vertex cover, maximum clique, minimum dominating set and graph coloring. In this work, a framework is proposed for solving problems of this type. The performance of this approach is explored for standard multi-layer GCNs using different types of convolutional layers. On top of this, we propose a new structure that is based on graph isomorphism networks. Computational experiments show that this new structure is significantly more efficient than basic structures. To further improve the performance of this method, the use of GCN ensembles is also explored. When using GCNs to generate solution values for GCOPs, they often correspond to non-feasible solutions because they violate the problem boundaries. This issue is addressed by defining an asymmetric loss function that is used during the GCN training. The proposed method is evaluated on a large set of training data consisting of graph solution pairs that can also be accessed online.

The Locating Chromatic Number for Certain Operation of Origami Graphs

The locating chromatic number introduced by Chartrand et al. in 2002 is the marriage of the partition dimension and graph coloring. The locating chromatic number depends on the minimum number of colors used in the locating coloring and the different color codes in vertices on the graph. There is no algorithm or theorem to determine the locating chromatic number of any graph carried out for each graph class or the resulting graph operation. This research is the development of scientific theory with a focus of the study on developing new ideas to determine the extent to which the locating chromatic number of a graph increases when applied to other operations. The locating chromatic number of the origami graph was obtained. The next exciting thing to know is locating chromatic number for certain operation of origami graphs. This paper discusses locating chromatic number for specific operation of origami graphs. The method used in this study is to determine the upper and lower bound of the locating chromatic number for certain operation of origami graphs. The result obtained is an increase of one color in the locating chromatic number of origami graphs.

Proper Colourings in r-Regular Zagreb Index Graph

Proper Colourings in r-Regular Zagreb Index Graph