Graph Coloring Research Articles

RAINBOW VERTEX CONNECTION NUMBER OF BULL GRAPH, NET GRAPH, TRIANGULAR LADDER GRAPH, AND COMPOSITION GRAPH (P_n [P_1

The rainbow connection was first introduced by Chartrand in 2006 and then in 2009 Krivelevich and Yuster first time introduced the rainbow vertex connection. Let graph be a connected graph. The rainbow vertex-connection is the assignment of color to the vertices of a graph , if every vertex on the graph is connected by a path graph that has interior vertices in different colors. The minimum number of colors from the rainbow vertex coloring in the graph is called rainbow vertex connection number which is denoted . The results of the research are the rainbow vertex connection number of bull graph, net graph, triangular ladder graph, and graph composition (Pn[P1]).

Defective acyclic colorings of planar graphs

AbstractThis paper studies two variants of defective acyclic coloring of planar graphs. For a graph and a coloring of , a 2‐colored cycle (2CC) transversal is a subset of that intersects every 2‐colored cycle. Let be a positive integer. We denote by the minimum integer such that has a proper ‐coloring which has a 2CC transversal of size , and by the minimum size of a subset of such that is acyclic ‐colorable. We prove that for any ‐vertex 3‐colorable planar graph and for any planar graph provided that . We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal can be chosen in such a way that induces a forest. We also prove that for any planar graph and .

Efficient verification of two-colorable graph states

Efficient verification of two-colorable graph states

Coloring Groups

We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.

Mixed Graph Colouring as Scheduling a Partially Ordered Set of Interruptible Multi-Processor Tasks with Integer Due Dates

We investigate relationships between scheduling problems with the bottleneck objective functions (minimising makespan or maximal lateness) and problems of optimal colourings of the mixed graphs. The investigated scheduling problems have integer durations of the multi-processor tasks (operations), integer release dates and integer due dates of the given jobs. In the studied scheduling problems, it is required to find an optimal schedule for processing the partially ordered operations, given that operation interruptions are allowed and indicated subsets of the unit-time operations must be processed simultaneously. First, we show that the input data for any considered scheduling problem can be completely determined by the corresponding mixed graph. Second, we prove that solvable scheduling problems can be reduced to problems of finding optimal colourings of corresponding mixed graphs. Third, finding an optimal colouring of the mixed graph is equivalent to the considered scheduling problem determined by the same mixed graph. Finally, due to the proven equivalence of the considered optimisation problems, most of the results that were proven for the optimal colourings of mixed graphs generate similar results for considered scheduling problems, and vice versa.

On (k, ℓ)-locating colorings of graphs

Let c: V(G) → {1, . , ℓ} = [ℓ] be a proper vertex coloring of G and C(i) = {u ∈ V(G): c(u) = i} for i ∈ [ℓ]. The k-color code rk (v|c) of vertex v is the ordered ℓ-tuple (aG (v,C(1)), . , aG (v,C(ℓ))) where If every two vertices have different color codes, then c is a (k, ℓ)-locating coloring of G. The k-locating chromatic number of graph G, denoted by , is the smallest integer ℓ such that G has a (k, ℓ)-locating coloring. In this paper, we propose this concept as an extension of diam(G)-locating chromatic number and 2-locating chromatic number which are known as the locating chromatic number, denoted χL (G), and neighbor-locating chromatic number, denoted , respectively. In this paper, we give sharp bounds for and where G◦H and are the corona and edge corona of G and H, respectively. We formulate an integer linear programming model to determine , noting that almost all graphs have diameter 2 and for every graph G of diameter 2.

A Comparative Study of Cat Swarm Algorithm for Graph Coloring Problem: Convergence Analysis and Performance Evaluation

The Graph Coloring Problem (GCP) is a significant optimization challenge widely suitable to solve scheduling problems. Its goal is to specify the minimum colors (k) required to color a graph properly. Due to its NP-completeness, exact algorithms become impractical for graphs exceeding 100 vertices. As a result, approximation algorithms have gained prominence for tackling large-scale instances. In this context, the Cat Swarm algorithm, a novel population-based metaheuristic in the domain of swarm intelligence, has demonstrated promising convergence properties compared to other population-based algorithms. This research focuses on designing and implementing the Cat Swarm algorithm to address the GCP. By conducting a comparative study with established algorithms, our investigation revolves around quantifying the minimum value of k required by the Cat Swarm algorithm for each graph instance. The evaluation metrics include the algorithm's running time in seconds, success rate, and the mean count of iterations or assessments required to reach a goal.

Distinguishing colorings, proper colorings, and covering properties without AC

Distinguishing colorings, proper colorings, and covering properties without AC

On Strong Regal and Strong Royal Colorings

Two colorings have been introduced recently where an unrestricted coloring c assigns nonempty subsets of [ k ] = { 1 , … , k } to the edges of a (connected) graph G and gives rise to a vertex-distinguishing vertex coloring by means of set operations. If each vertex color is obtained from the union of the incident edge colors, then c is referred to as a strong royal coloring. If each vertex color is obtained from the intersection of the incident edge colors, then c is referred to as a strong regal coloring. The minimum values of k for which a graph G has such colorings are referred to as the strong royal index of G and the strong regal index of G respectively. If the induced vertex coloring is neighbor distinguishing, then we refer to such edge colorings as royal and regal colorings. The royal chromatic number of a graph involves minimizing the number of vertex colors in an induced vertex coloring obtained from a royal coloring. In this paper, we provide new results related to these two coloring concepts and establish a connection between the corresponding chromatic parameters. In addition, we establish the royal chromatic number for paths and cycles.

Isolation of Regular Graphs and k-Chromatic Graphs

Given a set F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}$$\\end{document} of graphs, we call a copy of a graph in F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}$$\\end{document} an F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}$$\\end{document}-graph. The F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}$$\\end{document}-isolation number of a graph G, denoted by ι(G,F)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\iota (G,{\\mathcal {F}})$$\\end{document}, is the size of a smallest set D of vertices of G such that the closed neighborhood of D intersects the vertex sets of the F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}$$\\end{document}-graphs contained by G (equivalently, G-N[D]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G - N[D]$$\\end{document} contains no F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}$$\\end{document}-graph). Thus, ι(G,{K1})\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\iota (G,\\{K_1\\})$$\\end{document} is the domination number of G. For any integer k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ge 1$$\\end{document}, let F1,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{1,k}$$\\end{document} be the set of regular graphs of degree at least k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k-1$$\\end{document}, let F2,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{2,k}$$\\end{document} be the set of graphs whose chromatic number is at least k, and let F3,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{3,k}$$\\end{document} be the union of F1,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{1,k}$$\\end{document} and F2,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{2,k}$$\\end{document}. Thus, k-cliques are members of both F1,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{1,k}$$\\end{document} and F2,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{2,k}$$\\end{document}. We prove that for each i∈{1,2,3}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i \\in \\{1, 2, 3\\}$$\\end{document}, m+1k2+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{m+1}{{k \\atopwithdelims ()2} + 2}$$\\end{document} is a best possible upper bound on ι(G,Fi,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\iota (G, {\\mathcal {F}}_{i,k})$$\\end{document} for connected m-edge graphs G that are not k-cliques. The bound is attained by infinitely many (non-isomorphic) graphs. The proof of the bound depends on determining the graphs attaining the bound. This appears to be a new feature in the literature on isolation. Among the result’s consequences are a sharp bound of Fenech, Kaemawichanurat, and the present author on the k-clique isolation number and a sharp bound on the cycle isolation number.

Designing a Doctor's Practice Scheduling Application in a Hospital Using the Graph Coloring Method

Penjadwalan merupakan suatu kebutuhan yang umum pada saat ini. Salah satu contohnya adalah rumah sakit. Rumah sakit membutuhkan proses penjadwalan untuk menjadwalkan dokter yang akan bertugas dan menggunakan suatu ruangan. Hal ini disebabkan jumlah dokter yang banyak dan jumlah ruangan yang terbatas maka diperlukan sistem penjadwalan sesuai kebutuhan. Permasalahan yang terjadi di RSUD Petala Bumi, jumlah dokter yang bertugas di IGD cukup banyak. Namun terdapatnya kebingungan dari manajemen dalam mengatur jadwal dokter dipelayanan IGD. Penelitian ini merupakan pengembangan dari metode pewarnaan Graph yang dilakukan dalam penyusunan jadwal dokter di rumah sakit. Adapun penelitian ini dilakukan untuk mengatasi masalah penjadwalan yang sering tidak sesuai dan biasanya perlu diperbaiki berulang kali di rumah sakit. Metode yang digunakan untuk melakukan penjadwalan cukup beragam salah satunya yaitu metode pewarnaan Graph. Penelitian ini bertujuan untuk mengetahui penerapan pewarnaan graf dalam menyusun jadwal dokter. Penelitian ini merupakan jenis penelitian R&D dengan desain perancangan sistem dengan penerapan pewarnaan graf. Pewarnaan graf dilakukan untuk empat hari karena pada matriks pola pewarnaan graf mengalami pengulangan setelah hari keempat. Selama penyusunan empat hari, pewarnaan graf memiliki 10 warna atau bilangan kromatik dengan asumsi tiga shift dan dua dokter dalam satu shift. Oleh karena itu pengoptimalan penyusunan jadwal dokter pada IGD RSUD Petala Bumi minimal dapat dilakukan dengan 10 dokter.

Generalized Rainbow Differential Privacy

We study a new framework for designing differentially private (DP) mechanisms via randomized graph colorings, called rainbow differential privacy. In this framework, datasets are nodes in a graph, and two neighboring datasets are connected by an edge. Each dataset in the graph has a preferential ordering for the possible outputs of the mechanism, and these orderings are called rainbows. Different rainbows partition the graph of connected datasets into different regions. We show that if a DP mechanism at the boundary of such regions is fixed and it behaves identically for all same-rainbow boundary datasets, then a unique optimal $(\epsilon,\delta)$-DP mechanism exists (as long as the boundary condition is valid) and can be expressed in closed-form. Our proof technique is based on an interesting relationship between dominance ordering and DP, which applies to any finite number of colors and for $(\epsilon,\delta)$-DP, improving upon previous results that only apply to at most three colors and for $\epsilon$-DP. We justify the homogeneous boundary condition assumption by giving an example with non-homogeneous boundary condition, for which there exists no optimal DP mechanism.

Transitive coloring of graphs

Let V={V1,…,Vk} be a partition of the vertices of a graph G. We say that an automorphism f of G is a transitive color automorphism associate to V if there exist vertices vi∈Vi and vj∈Vj with 1≤i,j≤k, such that (i) f(vi)=vj, (ii) f(Vℓ)=Vℓ for ℓ with 1≤ℓ≤k and ℓ∉{i,j}. We denote f by fi,j. We say that V is transitive k-coloring if, for every two distinct colors i and j with 1≤i,j≤k, there exists a transitive color automorphism fi,j. The transitive coloring number of G, denoted by T(G), is the greatest integer k that G has a transitive k-coloring.The main purpose of this paper is to study some basic properties of transitive coloring graphs and determine the transitive coloring number of certain graphs.

Coloring of Non-Zero Component Graphs

Coloring of Non-Zero Component Graphs

Quantum No-signalling Correlations and Non-local Games

We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.

Compression algorithm for colored de Bruijn graphs

A colored de Bruijn graph (also called a set of k-mer sets), is a set of k-mers with every k-mer assigned a set of colors. Colored de Bruijn graphs are used in a variety of applications, including variant calling, genome assembly, and database search. However, their size has posed a scalability challenge to algorithm developers and users. There have been numerous indexing data structures proposed that allow to store the graph compactly while supporting fast query operations. However, disk compression algorithms, which do not need to support queries on the compressed data and can thus be more space-efficient, have received little attention. The dearth of specialized compression tools has been a detriment to tool developers, tool users, and reproducibility efforts. In this paper, we develop a new tool that compresses colored de Bruijn graphs to disk, building on previous ideas for compression of k-mer sets and indexing colored de Bruijn graphs. We test our tool, called ESS-color, on various datasets, including both sequencing data and whole genomes. ESS-color achieves better compression than all evaluated tools and all datasets, with no other tool able to consistently achieve less than 44% space overhead. The software is available at http://github.com/medvedevgroup/ESSColor.

Modification-fair cluster editing

The classic Cluster Editing problem (also known as Correlation Clustering) asks to transform a given graph into a disjoint union of cliques (clusters) by a small number of edge modifications. When applied to vertex-colored graphs (the colors representing subgroups), standard algorithms for the NP-hard Cluster Editing problem may yield solutions that are biased towards subgroups of data (e.g., demographic groups), measured in the number of modifications incident to the members of the subgroups. We propose a modification fairness constraint which ensures that the number of edits incident to each subgroup is proportional to its size. To start with, we study Modification-Fair Cluster Editing for graphs with two vertex colors. We show that the problem is NP-hard even if one may only insert edges within a subgroup; note that in the classic “non-fair” setting, this case is trivially polynomial-time solvable. However, in the more general editing form, the modification-fair variant remains fixed-parameter tractable with respect to the number of edge edits. We complement these and further theoretical results with an empirical analysis of our model on real-world social networks where we find that the price of modification-fairness is surprisingly low, that is, the cost of optimal modification-fair solutions differs from the cost of optimal “non-fair” solutions only by a small percentage.

Inclusive Local Irregularity Vertex Coloring In Grid Graph Family

Let is a simple graph and connected where is vertex set and is edge set. A maping as vertex k- labeling and function : is inclusive local irregularity vertex coloring, with . The minimum number of colors produced from inclusive local irregularity vertex coloring of graph is called inclusive chromatic number local irregularity, denoted by . On this paper, we learn about the inclusive local irregularity vertex coloring and determine the chromatic number on grid graph family.

On strong incidence coloring of subcubic graphs

An incidence of a graph [Formula: see text] is a pair [Formula: see text] where [Formula: see text] is a vertex of [Formula: see text] and [Formula: see text] is an edge of [Formula: see text] incident with [Formula: see text]. Two incidences [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent whenever (i) [Formula: see text], or (ii) [Formula: see text] or (iii) [Formula: see text] or [Formula: see text]. A strong incidence coloring of a graph [Formula: see text] is a mapping from the set of incidences of [Formula: see text] to the set of colors [Formula: see text], such that every two incidences that are adjacent, or adjacent to the same incidence receive distinct colors. In this paper, we prove that every connected subcubic graph [Formula: see text] except [Formula: see text] has a strong incidence coloring with at most [Formula: see text] colors.

2-Distance coloring of planar graphs with maximum degree 5

A 2-distance coloring of a graph is a proper [Formula: see text]-coloring in which any two vertices with distance at most two get different colors. The 2-distance number is the smallest number [Formula: see text] such that [Formula: see text] has a 2-distance [Formula: see text]-coloring, denoted as [Formula: see text]. In 1977, Wegner conjectured that for each planar graph [Formula: see text] with maximum degree [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text] if [Formula: see text]. In 2001, Thomassen supported the conjecture by proving the case [Formula: see text]. The conjecture is still open even for [Formula: see text]. In this paper, we show that [Formula: see text] for the case [Formula: see text] which improves the upper bound 18 recently obtained by Hou et al.