Graph Coloring Research Articles

The Influence of Graph Theory Approach in Preventing Cheating at Higher Learning Institution: A case of Institute of Accountancy Arusha

Academic dishonesty has been more common in recent years, both through self-reported and detected cases, which has brought significant attention to the issue of cheating in educational settings. Students often cheat for a variety of reasons, including lack of study time, unclear learning objectives, fear of failing, peer pressure, and parental pressure for academic success. Students who rely on plagiarizing answers miss the opportunity to build their skills and may face challenges in job requirements. The problem of cheating / copying the neighbor's works during the examination is particularly well known. With graph theory especially graph coloring, it is proposed to develop the students' sitting arrangement that can be the solution of cheating. An experimental case study focuses on three master's programs, three bachelor's degree programs, three diploma programs, and three certificate programs at the Institute of Accountancy Arusha. This is done when one room or a venue is assigned to two or more programs with different backgrounds depending on the size of the room. The result shows that using graph coloring has reduced the tendency of students to copy or ask questions from their neighbors in the examination room. Based on the assessment of graph theory's effectiveness in preventing cheating at the Institute of Accountancy Arusha, several recommendations can be made. First, the institution should invest in developing a comprehensive data collection system that captures relevant information such as seating arrangements, student interactions, and exam performance patterns. Collaboration with other institutions in implementing similar technologies could also create a shared knowledge base and lead to best practices in academic integrity. Finally, continuous monitoring and evaluation of this system should be conducted to refine its processes and adapt to new cheating technique.

On Radio k – Chromatic Number for Mycielski of Some Graphs

Objectives: In radio networks, the main difficulty is managing the radio spectrum, assigning radio frequencies to transmitters optimally without any interferences. This study aims to find the smallest span of Mycielski of some graphs, the maximum color assigned to any node is called span. Methods : This study focused on the problem of reducing interference by modeling it with a radio k - coloring problem on graphs. Where transmitters are modeled as nodes in a graph, with edges connecting nodes that represent transmitters in close proximity to one another. For a graph , with node set , edge set and an integer , a radio k - coloring of is a function satisfying the condition for any two nodes and , where is the distance between and in . The radio k - coloring of , is the maximum color assigned to any node of and it is denoted by . The radio k -chromatic number of is the minimum value of taken over all radio k - coloring of and it is denoted by . Findings: This study obtained the radio k-chromatic number for Mycielski of some graphs for and 3 Novelty: To solve the channel assignment problem in radio transmitters, the interference graph is developed, and the channel assignment has been converted into a graph coloring. Reducing the interference by a radio k - coloring problem will motivate many researchers to find the radio k - coloring in various graphs. Keywords: Radio k – Coloring, Mycielski graph, Double Star graph, Triple Star graph, Sunlet graph, Helm graph and Closed Helm graph

Online coloring of disk graphs

In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of disks with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of them is an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online L(2,1)-labeling.

Flip colouring of graphs

It is proved that for integers b, r such that 3≤b<r≤b+12-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3 \\le b < r \\le \\left( {\\begin{array}{c}b+1\\\\ 2\\end{array}}\\right) - 1$$\\end{document}, there exists a red/blue edge-colored graph such that the red degree of every vertex is r, the blue degree of every vertex is b, yet in the closed neighbourhood of every vertex there are more blue edges than red edges. The upper bound r≤b+12-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\le \\left( {\\begin{array}{c}b+1\\\\ 2\\end{array}}\\right) -1$$\\end{document} is best possible for any b≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b \\ge 3$$\\end{document}. We further extend this theorem to more than two colours, and to larger neighbourhoods. A useful result required in some of our proofs, of independent interest, is that for integers r, t such that 0≤t≤r22-5r3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le t \\le \\frac{r^2}{2} - 5r^{3/2}$$\\end{document}, there exists an r-regular graph in which each open neighbourhood induces precisely t edges. Several explicit constructions are introduced and relationships with constant linked graphs, (r, b)-regular graphs and vertex transitive graphs are revealed.

Perfect Resolution of Strong Conflict-Free Colouring of Interval Hypergraphs

The \(k\) -Strong Conflict-Free ( \(k\) -SCF, in short) colouring problem seeks to find a colouring of the vertices of a hypergraph \(H\) using minimum number of colours so that in every hyperedge \(e\) of \(H\) , there are at least \(\min\{|e|,k\}\) vertices whose colours are different from that of all other vertices in \(e\) . In the case of interval hypergraphs, we present an exact \({\mathsf{P}}\) -time algorithm for the \(k\) -SCF problem thus solving an open problem posed by Cheilaris et al. (2014). We achieve our results by showing that for any hypergraph, a \(k\) -SCF colouring is a proper colouring of a related simple graph which we refer to as a co-occurrence graph . We then show that a co-occurrence graph is obtained by identifying an induced subgraph of a second simple graph that we introduce, which we refer to as the conflict graph . For interval hypergraphs, we show that each co-occurrence graph and the conflict graph are perfect graphs. This property plays a crucial role in our polynomial time algorithm. Secondly, we show that for an interval hypergraph, the \(1\) -SCF colouring number is the minimum partition of its intervals into sets such that each set has an exact hitting set (a hitting set in which each interval is hit exactly once).

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

Characterizations and Clique Coloring of Edge Intersection Graphs on a Triangular Grid

Interpreting the Projected Frontal Area in Front Crawl: Determining the Projected Frontal Area of Each Body Segment.

This study aimed to provide evidence for the interpretation of the projected frontal area (PFA) during front crawl. To achieve this goal, we developed a method for calculating the PFA of each body segment using digital human technology and compared the pressure drag under two calculation conditions: a combination of the PFA with and without accounting for the horizontal velocity of each body segment. Twelve competitive male swimmers performed a 15-meter front crawl at 1.20 m·s-1. The three-dimensional positions of the reflective markers attached to the swimmer's body were recorded using an underwater motion-capture system. Based on the body shape of each swimmer obtained from the photogenic body scanner, individual digital human body models were created with the color of the model's vertices divided into eight body segments. The time series of the volumetric swimming motion was reconstructed using inverse kinematics. The PFA of each body segment was then calculated by the automatic processing of a series of parallel frontal images. The pressure drag index, defined as the value excluding the drag coefficient while simultaneously considering the PFA and the horizontal velocity, was calculated under two conditions: the static condition (accounting for only the PFA of each body segment) and the dynamic condition (accounting for the PFA and horizontal velocity of each body segment). Notably, the pressure drag index was higher under the static condition than under the dynamic condition for the humerus, ulna, and hand segments (p < 0.001). The results obtained using our methodology indicate that the PFA of the upper limb segments overestimates their contribution to pressure drag during front crawl under the static condition.

Chromatic Coloring of Distance Graphs V

Let G = ( V , E ) be any graph. If there exists an injection f : V → Z , such that | f ( u ) – f ( v ) | is prime for every u v ∈ E , then we say G is a prime distance graph (PDG). The problem of characterizing the family of all prime distance graphs (PDGs) with chromatic number 3 or 4 is challenging. In the fourth part of this series of articles, we determined which fans are PDGs and which wheels are PDGs. In addition, we showed: (1) a chain of n mutually isomorphic PDGs is a PDG, and (2) the Cartesian product of a PDG and a path is a PDG. In this part of the series, we improve (1) by showing that there exists a chain of n arbitrary PDGs which is a PDG. We also show that the following graphs are PDGs: (a) any graph with at most three cycles, (b) the one-point union of cycles, and (c) a family of graphs consisting of paths with common end vertices.

Optimality of DSatur algorithm on chordal graphs

DSatur is a widely-used heuristic algorithm for graph coloring, proposed by Daniel Brélaz in 1979. It is based on the greedy coloring approach, but selecting the next vertex to be colored from those that maximize the number of colors already used by their neighbors. Though not always optimal, this algorithm is known to be optimal on certain families, like cycles or bipartite graphs. In this paper, we prove the optimality of DSatur on chordal graphs, a superclass of interval graphs.

Neighbor Locating Coloring on Graphs: Three Products

Let G be a simple finite graph. A k -coloring of G is a partition π = { S 1 , ⋯ , S k } of V ( G ) so that each S i is an independent set and any vertex in S i takes color i . A k -coloring π = { S 1 , ⋯ , S k } of V ( G ) is a neighbor locating coloring if for any two vertices u , v ∈ S i , there is a color class S j for which one of them has a neighbor in S j and the other does not. The minimum k with this property is said to be the neighbor locating chromatic number of G , denoted by χ N L ( G ) . We initiate the study of the neighbor locating coloring of graphs resulting from three types of product of two graphs. We investigate the neighbor locating chromatic number of Cartesian, lexicographic, and corona products of two graphs. Finally, we untangle the neighbor locating chromatic number of any of the aforementioned three products of cycles, paths, and complete graphs.

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

Structural Parameterizations for Two Bounded Degree Problems Revisited

We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring , where the input is a graph G and a target degree Δ, and we are asked either to edit or partition the graph so that the maximum degree becomes bounded by Δ. Both problems are known to be parameterized intractable for the most well-known structural parameters, such as treewidth. We revisit the parameterization by treewidth, as well as several related parameters and present a more fine-grained picture of the complexity of both problems. In particular, we present the following: — Both problems admit straightforward DP algorithms with table sizes \((\Delta +2)^\mathrm{tw}\) and \((\chi _\mathrm{d}(\Delta +1))^{\mathrm{tw}}\) , respectively, where tw is the input graph’s treewidth and \(\chi _\mathrm{d}\) the number of available colors. We show that, under the SETH, both algorithms are essentially optimal, for any non-trivial fixed values of \(\Delta , \chi _\mathrm{d}\) , even if we replace treewidth by pathwidth. Along the way, we obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for treewidth and pathwidth. — Given that the standard DP algorithm is optimal for treewidth and pathwidth, we then go on to consider the more restricted parameter tree-depth. Here, previously known lower bounds imply that, under the ETH, Bounded Vertex Degree Deletion and Defective Coloring cannot be solved in time \(n^{o(\sqrt [4]{\mathrm{td}})}\) and \(n^{o(\sqrt {\mathrm{td}})}\) , respectively, leaving some hope that a qualitatively faster algorithm than the one for treewidth may be possible. We close this gap by showing that neither problem can be solved in time \(n^{o(\mathrm{td})}\) , under the ETH, by employing a recursive low tree-depth construction that may be of independent interest. — Finally, we consider a structural parameter that is known to be restrictive enough to render both problems FPT: vertex cover. For both problems the best known algorithm in this setting has a super-exponential dependence of the form \(\mathrm{vc}^{\mathcal {O}(\mathrm{vc})}\) . We show that this is optimal, as an algorithm with dependence of the form \(\mathrm{vc}^{o(\mathrm{vc})}\) would violate the ETH. Our proof relies on a new application of the technique of d -detecting families introduced by Bonamy et al. [ToCT 2019]. Our results, although mostly negative in nature, paint a clear picture regarding the complexity of both problems in the landscape of parameterized complexity, since in all cases we provide essentially matching upper and lower bounds.

On Total Coloring of Triple Star and Lobster Graphs

On Total Coloring of Triple Star and Lobster Graphs

Colored vertex models and Iwahori Whittaker functions

We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group G over a nonarchimedean local field F. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case G=GLr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=\ extrm{GL}_r$$\\end{document} we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here ‘solvable’ means that the models have a family of Yang–Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang–Baxter equations come from a Drinfeld twist of the quantum group Uq(gl^(r|1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_q(\\widehat{\\mathfrak {gl}}(r|1))$$\\end{document}, which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.

Achromatic colorings of polarity graphs

A complete partition of a graph G is a partition of the vertex set such that there is at least one edge between any two parts. The largest r such that G has a complete partition into r parts, each of which is an independent set, is the achromatic number of G. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of C4-free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.

On the Complexity of Local-Equitable Coloring in Claw-Free Graphs with Small Degree

On the Complexity of Local-Equitable Coloring in Claw-Free Graphs with Small Degree

On non-superperfection of edge intersection graphs of paths

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

Maximal Colourings for Graphs

We consider two different notions of graph colouring, namely, the t-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them.

Proper edge colorings of planar graphs with rainbow C4 ${C}_{4}$‐s

AbstractWe call a proper edge coloring of a graph a B‐coloring if every 4‐cycle of is colored with four different colors. Let denote the smallest number of colors needed for a B‐coloring of . Motivated by earlier papers on B‐colorings, here we consider for planar and outerplanar graphs in terms of the maximum degree . We prove that for planar graphs, for bipartite planar graphs, and for outerplanar graphs with . We conjecture that, for sufficiently large, for planar , and for outerplanar .

Local inclusive distance antimagic coloring of graphs

For a simple graph &lt;em&gt;G&lt;/em&gt;, a bijection &lt;em&gt;f&lt;/em&gt; : &lt;em&gt;V&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) → [1,|&lt;em&gt;V&lt;/em&gt; (&lt;em&gt;G&lt;/em&gt;)|] is called as a &lt;em&gt;local inclusive distance antimagic&lt;/em&gt; (&lt;em&gt;LIDA&lt;/em&gt;) &lt;em&gt;labeling&lt;/em&gt; of &lt;em&gt;G&lt;/em&gt; if &lt;em&gt;w&lt;/em&gt;(&lt;em&gt;u&lt;/em&gt;) ≠ &lt;em&gt;w&lt;/em&gt;(&lt;em&gt;v&lt;/em&gt;) for every two adjacent vertices &lt;em&gt;u,v&lt;/em&gt; ∈ &lt;em&gt;V&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) with &lt;em&gt;w&lt;/em&gt;(&lt;em&gt;u&lt;/em&gt;) = ∑&lt;sub&gt;&lt;em&gt;x&lt;/em&gt;∈&lt;em&gt;N&lt;/em&gt; [&lt;em&gt;u&lt;/em&gt;]&lt;/sub&gt; &lt;em&gt;f&lt;/em&gt;(&lt;em&gt;x&lt;/em&gt;). A graph &lt;em&gt;G&lt;/em&gt; is said to be local inclusive distance antimagic (LIDA) graph if it admits a LIDA labeling. The function &lt;em&gt;w&lt;/em&gt; induced by &lt;em&gt;f&lt;/em&gt; also can be seen as a proper vertex coloring of &lt;em&gt;G&lt;/em&gt;. The &lt;em&gt;local inclusive distance antimagic&lt;/em&gt; (&lt;em&gt;LIDA&lt;/em&gt;) &lt;em&gt;chromatic number&lt;/em&gt; of &lt;em&gt;G&lt;/em&gt;, denoted by χ&lt;sub&gt;&lt;em&gt;lida&lt;/em&gt;&lt;/sub&gt;(&lt;em&gt;G&lt;/em&gt;), is the minimum number of colors taken over all proper vertex colorings induced by LIDA labelings of &lt;em&gt;G&lt;/em&gt;. In this paper, we study a LIDA labeling of simple graph. We provide some basic properties of LIDA labeling for any simple graphs. The LIDA chromatic number of certain multipartite graphs, double stars, subdivision of graphs and join product of graphs with &lt;em&gt;K&lt;/em&gt;&lt;sub&gt;1&lt;/sub&gt; are also investigated. We present an upper bound for graphs obtained from subdivision of super edge-magic total graphs. Furthermore, we present some new open problems.