Special Issue "Graph-Theoretic Problems and Their New Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 January 2020).

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Special Issue Editor

Prof. Dr. Frank Werner
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Special Issue Information

Dear Colleagues,

Nowadays, graph theory plays a vital role in many disciplines. We invite you to submit your latest original research results in all aspects of graph theory to this Special Issue. We are looking both for new and innovative approaches for traditional graph-theoretic problems and well as for new applications of graph theory in emerging fields, such as in network security, computer science and data analysis, bioinformatics, operations research, engineering and manufacturing, physics and chemistry, linguistics, or social sciences. Both theoretical results, as well as new graph algorithms, with investigations of their computational complexity will be considered. Below, we give an exemplary, but not exhaustive, list of potential subjects for submissions to this Special Issue.

Prof. Dr. Frank Werner
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Deterministic, randomized, exact and approximative graph algorithms
  • Paths, cycles, and trees
  • Network theory
  • Graph labeling
  • Graph coloring
  • Symmetric graphs
  • Polyhedral graphs
  • Topological indices
  • Domination in graphs
  • Applications of graph theory

Published Papers (21 papers)

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Editorial

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Open AccessEditorial
Graph-Theoretic Problems and Their New Applications
Mathematics 2020, 8(3), 445; https://doi.org/10.3390/math8030445 - 19 Mar 2020
Abstract
Graph theory is an important area of Applied Mathematics with a broad spectrum of applications in many fields [...] Full article

Research

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Open AccessArticle
Competition-Independence Game and Domination Game
Mathematics 2020, 8(3), 359; https://doi.org/10.3390/math8030359 - 05 Mar 2020
Cited by 1
Abstract
The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as [...] Read more.
The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as many turns as possible. The game ends when all vertices are dominated. The game domination number, denoted by γ g ( G ) (respectively γ g ( G ) ), is the total number of turns when both players play optimally and when Dominator (respectively Staller) starts the game. In this paper, we study a version of this game where the set of chosen vertices is always independent. This version turns out to be another game known as the competition-independence game. The competition-independence game is played on a graph by two players, Diminisher and Sweller. They take turns in constructing maximal independent set M, where Diminisher tries to minimize | M | and Sweller tries to maximize | M | . Note that, actually, it is the domination game in which the set of played vertices is independent. The competition-independence number, denoted by I d ( G ) (respectively I s ( G ) ) is the optimal size of the final independent set in the competition-independence game if Diminisher (respectively Sweller) starts the game. In this paper, we check whether some well-known results in the domination game hold for the competition-independence game. We compare the competition-independence numbers to the game domination numbers. Moreover, we provide a family of graphs such that many parameters are equal. Finally, we present a realization result on the competition-independence numbers. Full article
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Open AccessArticle
Complexity Analysis and Stochastic Convergence of Some Well-known Evolutionary Operators for Solving Graph Coloring Problem
Mathematics 2020, 8(3), 303; https://doi.org/10.3390/math8030303 - 25 Feb 2020
Cited by 1
Abstract
The graph coloring problem is an NP-hard combinatorial optimization problem and can be applied to various engineering applications. The chromatic number of a graph G is defined as the minimum number of colors required to color the vertex set V(G [...] Read more.
The graph coloring problem is an NP-hard combinatorial optimization problem and can be applied to various engineering applications. The chromatic number of a graph G is defined as the minimum number of colors required to color the vertex set V(G) so that no two adjacent vertices are of the same color, and different approximations and evolutionary methods can find it. The present paper focused on the asymptotic analysis of some well-known and recent evolutionary operators for finding the chromatic number. The asymptotic analysis of different crossover and mutation operators helps in choosing the better evolutionary operator to minimize the problem search space and computational complexity. The choice of the right genetic operators facilitates an evolutionary algorithm to achieve faster convergence with lesser population size N through an adequate distribution of promising genes. The selection of an evolutionary operator plays an essential role in reducing the bounds for minimum color obtained so far for some of the benchmark graphs. This research also focuses on the necessary and sufficient conditions for the global convergence of evolutionary algorithms. The stochastic convergence of recent evolutionary operators for solving graph coloring is newly analyzed. Full article
Open AccessArticle
Some Results on (sq)-Graphic Contraction Mappings in b-Metric-Like Spaces
Mathematics 2019, 7(12), 1190; https://doi.org/10.3390/math7121190 - 04 Dec 2019
Cited by 2
Abstract
In this paper we consider (sq)-graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve [...] Read more.
In this paper we consider ( s q ) -graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results. Full article
Open AccessArticle
f-Polynomial on Some Graph Operations
Mathematics 2019, 7(11), 1074; https://doi.org/10.3390/math7111074 - 08 Nov 2019
Cited by 1
Abstract
Given any function f:Z+R+, let us define the f-index If(G)=uV(G)f(du) and the f-polynomial Pf(G [...] Read more.
Given any function f : Z + R + , let us define the f-index I f ( G ) = u V ( G ) f ( d u ) and the f-polynomial P f ( G , x ) = u V ( G ) x 1 / f ( d u ) 1 , for x > 0 . In addition, we define P f ( G , 0 ) = lim x 0 + P f ( G , x ) . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others. Full article
Open AccessArticle
An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics
Mathematics 2019, 7(11), 1068; https://doi.org/10.3390/math7111068 - 06 Nov 2019
Cited by 1
Abstract
Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any [...] Read more.
Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any isotopism of the former is uniquely related to an isomorphism of the latter. This enables us to develop some results on graph theory in the context of the molecular processes that occur during the S-phase of a mitotic cell cycle. In particular, each monochromatic subset of edges is identified with a mutation or regulatory mechanism that relates any two statuses of the genotypes of a pair of chromatids. Full article
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Open AccessArticle
Matching Extendabilities of G = CmPn
Mathematics 2019, 7(10), 941; https://doi.org/10.3390/math7100941 - 11 Oct 2019
Cited by 1
Abstract
A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting G [...] Read more.
A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting G = C m P n ( m 3 and n 1 ) be the graph join of C m (the cycle with m vertices) and P n (the path with n vertices) contains a perfect matching, we find necessary and sufficient conditions for G to be induced-matching extendable and bipartite-matching extendable. Full article
Open AccessArticle
A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
Mathematics 2019, 7(6), 551; https://doi.org/10.3390/math7060551 - 17 Jun 2019
Cited by 5
Abstract
Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy [...] Read more.
Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the d m - and t d m -degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely d m -regular, t d m -regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the μ -complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here. Full article
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Open AccessArticle
On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution
Mathematics 2019, 7(5), 472; https://doi.org/10.3390/math7050472 - 24 May 2019
Cited by 1
Abstract
The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree [...] Read more.
The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree problems of fan graphs, wheel graphs, and the class of graphs obtained from “partitioning” wheel graphs under dynamic evolution. The enumeration of these subtree numbers is done through the so-called subtree generation functions of graphs. With the enumerative result, we briefly explore the extremal problems in the corresponding class of graphs. Some interesting observations on the behavior of the subtree number are also presented. Full article
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Open AccessArticle
Wirelength of Enhanced Hypercube into Windmill and Necklace Graphs
Mathematics 2019, 7(5), 383; https://doi.org/10.3390/math7050383 - 26 Apr 2019
Cited by 1
Abstract
An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength [...] Read more.
An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength problem reduces time and cost in the embedded design. Due to the potential topological properties of enhanced hypercube, it has become constructive in recent years, and a lot of research work has been carried out on it. In this paper, we use the edge isoperimetric problem to produce the exact wirelengths of embedding enhanced hypercube into windmill and necklace graphs. Full article
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Open AccessArticle
Reformulated Zagreb Indices of Some Derived Graphs
Mathematics 2019, 7(4), 366; https://doi.org/10.3390/math7040366 - 22 Apr 2019
Cited by 4
Abstract
A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. [...] Read more.
A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph. Full article
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Open AccessArticle
The Bounds of Vertex Padmakar–Ivan Index on k-Trees
Mathematics 2019, 7(4), 324; https://doi.org/10.3390/math7040324 - 01 Apr 2019
Cited by 10
Abstract
The Padmakar–Ivan (PI) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges uv of a graph such that these vertices are not equidistant from u [...] Read more.
The Padmakar–Ivan ( P I ) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges u v of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of P I -indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the P I -values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions. Full article
Open AccessArticle
On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks
Mathematics 2019, 7(4), 314; https://doi.org/10.3390/math7040314 - 28 Mar 2019
Cited by 2
Abstract
The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let Hn be the linear heptagonal networks. It is interesting to [...] Read more.
The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n . Full article
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Open AccessArticle
Distance Degree Index of Some Derived Graphs
Mathematics 2019, 7(3), 283; https://doi.org/10.3390/math7030283 - 19 Mar 2019
Cited by 2
Abstract
Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (DD) index was introduced by Dobrynin and Kochetova in 1994 for characterizing [...] Read more.
Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree ( D D ) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a D D index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the D D index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph. Full article
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Open AccessArticle
More Results on the Domination Number of Cartesian Product of Two Directed Cycles
Mathematics 2019, 7(2), 210; https://doi.org/10.3390/math7020210 - 24 Feb 2019
Cited by 4
Abstract
Let γ(D) denote the domination number of a digraph D and let CmCn denote the Cartesian product of Cm and Cn, the directed cycles of length nm3. Liu [...] Read more.
Let γ ( D ) denote the domination number of a digraph D and let C m C n denote the Cartesian product of C m and C n , the directed cycles of length n m 3 . Liu et al. obtained the exact values of γ ( C m C n ) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ ( C m C n ) for m = 6 , 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ ( C m C n ) for m = 3 k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C m C n with m up to 21. Moreover, the exact values of γ ( C n C n ) with n up to 31 are determined. Full article
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Open AccessArticle
k-Rainbow Domination Number of P3Pn
Mathematics 2019, 7(2), 203; https://doi.org/10.3390/math7020203 - 21 Feb 2019
Cited by 2
Abstract
Let k be a positive integer, and set [k]:={1,2,,k}. For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping f:V [...] Read more.
Let k be a positive integer, and set [ k ] : = { 1 , 2 , , k } . For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping f : V ( G ) 2 [ k ] in such a way that, for any vertex v V ( G ) with the empty set under f, the condition u N G ( v ) f ( u ) = [ k ] always holds, where N G ( v ) is the open neighborhood of v. The weight of kRDF f of G is the summation of values of all vertices under f. The k-rainbow domination number of G, denoted by γ r k ( G ) , is the minimum weight of a kRDF of G. In this paper, we obtain the k-rainbow domination number of grid P 3 P n for k { 2 , 3 , 4 } . Full article
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Open AccessArticle
Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs
Mathematics 2019, 7(1), 72; https://doi.org/10.3390/math7010072 - 10 Jan 2019
Cited by 6
Abstract
Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of [...] Read more.
Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined; i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work. Full article
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Open AccessArticle
The Aα-Spectral Radii of Graphs with Given Connectivity
Mathematics 2019, 7(1), 44; https://doi.org/10.3390/math7010044 - 04 Jan 2019
Cited by 2
Abstract
The Aα-matrix is Aα(G)=αD(G)+(1α)A(G) with α[0,1], given by Nikiforov in 2017, where A( [...] Read more.
The A α -matrix is A α ( G ) = α D ( G ) + ( 1 α ) A ( G ) with α [ 0 , 1 ] , given by Nikiforov in 2017, where A ( G ) is adjacent matrix, and D ( G ) is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of A α ( G ) is said to be the A α -spectral radius of G. In this work, we determine the graphs with largest A α ( G ) -spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying A α ( G ) -spectral radius are proposed. Full article
Open AccessArticle
The Bounds of the Edge Number in Generalized Hypertrees
Mathematics 2019, 7(1), 2; https://doi.org/10.3390/math7010002 - 20 Dec 2018
Cited by 1
Abstract
A hypergraph H=(V,ε) is a pair consisting of a vertex set V, and a set ε of subsets (the hyperedges of H) of V. A hypergraph H is r-uniform if all the hyperedges [...] Read more.
A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T E is a fixed value k   ( 2 k r ) . We focus on the case that G H T E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n r + 1 edges if r 3   and   n 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T E has exactly k   ( 2 k r 1 ) components. Full article
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Open AccessArticle
Kempe-Locking Configurations
Mathematics 2018, 6(12), 309; https://doi.org/10.3390/math6120309 - 07 Dec 2018
Cited by 1
Abstract
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored [...] Read more.
The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored in this article is that the connectivity and coloring properties are incompatible. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. All Kempe-locked triangulations that we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say x y , and (2) they have a Birkhoff diamond with endpoints x and y as a subgraph. On the strength of our investigations, we formulate a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample are indeed incompatible. It would also imply the appealing conclusion that the Birkhoff diamond configuration alone is responsible for the 4-colorability of planar triangulations. Full article
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Review

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Open AccessReview
Mixed Graph Colorings: A Historical Review
Mathematics 2020, 8(3), 385; https://doi.org/10.3390/math8030385 - 09 Mar 2020
Cited by 1
Abstract
This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the [...] Read more.
This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the vertices of the mixed graph and one coloring of the arcs and edges of the mixed graph have been considered in the literature. The unit-time scheduling problem with the makespan criterion may be interpreted as an optimal coloring of the vertices of a mixed graph, where the number of used colors is minimum. Complexity results for optimal colorings of the mixed graph are systematized. The published algorithms for finding optimal mixed graph colorings are briefly surveyed. Two new colorings of a mixed graph are introduced. Full article
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