Academic literature on the topic 'Edge-colored graph theory'

Max-Cut problem is one of the classical problems in graph theory and has been widely studied in recent years. Maximum colored cut problem is a more general problem, which is to find a bipartition of a given edge-colored graph maximizing the number of colors in edges going across the bipartition. In this work, we gave some lower bounds on maximum colored cuts in edge-colored complete graphs containing no rainbow triangles or properly colored 4-cycles.

Abstract A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

The article proposes the results of the search for minimal vertex extensions of undirected colored complete graphs. The research topic is related to the modelling of full fault tolerant technical systems with a different type of their objects in the terminology of graph theory. Let a technical system be Σ, then there is a graph G(Σ), which vertices reflects system’s objects and edges reflects connections between these objects. Type of each object reflected in a mapping of some color from F = {1,2…,i} to the corresponding vertex. System’s Σ vertex extension is a graph G(Σ) which contains additional vertices. System reflected by graph G(Σ) can work even if there are k faults of its objects. Complete graph is a graph where each two vertices have an edge between them. Complete graphs have no edge extensions because there is no way to add additional edge to the graph with a maximum number of edges. In other words, the system reflected by some complete graph cannot be able to resist connection faults. Therefore the article research is focused on vertex extensions only. There is a description of vertex extensions existence condition for those colored complete graphs. This paper considers generating schemes for such minimal vertex extensions along with formulas, which allows to calculate number of additional edges to have an ability to construct minimal vertex extension.

Many symmetric properties are well-explored in graph theory, especially in graph coloring, such as symmetric graphs defined by the automorphism groups, symmetric drawing of planar graphs, and symmetric functions which are used to count the number of specific colorings of a graph. This paper is devoted to studying the star edge coloring of 1-planar graphs. The star chromatic index χst′(G) of a graph G is defined as the smallest k for which the edges of G can be colored by using k colors so that no two adjacent edges get the same color and no bichromatic paths or cycles of length four are produced. A graph G is called 1-planar if it can be drawn in the plane such that each edge crosses at most one other edge. In this paper, we prove that every 1-planar graph G satisfies χst′(G)≤7.75Δ+166; and moreover χst′(G)≤⌊1.5Δ⌋+500 if G contains no 4-cycles, and χst′(G)≤2.75Δ+116 if G is 3-connected, or optimal, or NIC-planar.

A strong edge coloring of a graph G is a proper coloring of edges in G such that any two edges of distance at most 2 are colored with distinct colors. The strong chromatic index χs′(G) is the smallest integer l such that G admits a strong edge coloring using l colors. A K4(t)-minor free graph is a graph that does not contain K4(t) as a contraction subgraph, where K4(t) is obtained from a K4 by subdividing edges exactly t−4 times. The paper shows that every K4(t)-minor free graph with maximum degree Δ(G) has χs′(G)≤(t−1)Δ(G) for t∈{5,6,7} which generalizes some known results on K4-minor free graphs by Batenburg, Joannis de Verclos, Kang, Pirot in 2022 and Wang, Wang, and Wang in 2018. These upper bounds are sharp.

A rectangular partition is a partition of a rectangle into non-overlapping rectangles, such that no four rectangles meet at a common point. A vertex guard is a guard located at a vertex of the partition (i.e., at a corner of a rectangle); it guards the rectangles that meet at this vertex. An edge guard is a guard that patrols along an edge of the partition, and is thus equivalent to two adjacent vertex guards. We consider the problem of finding a minimum-cardinality guarding set for the rectangles of the partition. For vertex guards, we prove that guarding a given subset of the rectangles is NP-hard. For edge guards, we prove that guarding all rectangles, where guards are restricted to a given subset of the edges, is NP-hard. For both results we show a reduction from vertex cover in non-bipartite 3-connected cubic planar graphs of girth greater than three. For the second NP-hardness result, we obtain a graph-theoretic result which establishes a connection between the set of faces of a plane graph of vertex degree at most three and a vertex cover for this graph. More precisely, we prove that one can assign to each internal face a distinct vertex of the cover, which lies on the face's boundary. We show that the vertices of a rectangular partition can be colored red, green, or black, such that each rectangle has all three colors on its boundary. We conjecture that the above is also true for four colors. Finally, we obtain a worst-case upper bound on the number of edge guards that are sufficient for guarding rectangular partitions with some restrictions on their structure.

RNA tertiary structure is crucial to its many non-coding molecular functions. RNA architecture is shaped by its secondary structure composed of stems, stacked canonical base pairs, enclosing loops. While stems are precisely captured by free-energy models, loops composed of non-canonical base pairs are not. Nor are distant interactions linking together those secondary structure elements (SSEs). Databases of conserved 3D geometries (a.k.a. modules) not captured by energetic models are leveraged for structure prediction and design, but the computational complexity has limited their study to local elements, loops. Representing the RNA structure as a graph has recently allowed to expend this work to pairs of SSEs, uncovering a hierarchical organization of these 3D modules, at great computational cost. Systematically capturing recurrent patterns on a large scale is a main challenge in the study of RNA structures. In this paper, we present an efficient algorithm to compute maximal isomorphisms in edge colored graphs. We extend this algorithm to a framework well suited to identify RNA modules, and fast enough to considerably generalize previous approaches. To exhibit the versatility of our framework, we first reproduce results identifying all common modules spanning more than 2 SSEs, in a few hours instead of weeks. The efficiency of our new algorithm is demonstrated by computing the maximal modules between any pair of entire RNA in the non-redundant corpus of known RNA 3D structures. We observe that the biggest modules our method uncovers compose large shared sub-structure spanning hundreds of nucleotides and base pairs between the ribosomes of Thermus thermophilus, Escherichia Coli, and Pseudomonas aeruginosa.

At the beginning of each semester, subjects scheduling is always carried out by the curriculum representatives and academic staff. There were several problems that must be avoided in subjects scheduling, these problems were the schedule of teachers who teach one subject at the same time are scheduled in different classes, teachers who teach more than one subject are scheduled in the same class at the same time, teachers who are lack of scheduled for teaching. In the subject of graph theory, there is a concept of graph coloring, one of which is vertex coloring. In vertex coloring, there is a Welch-Powell Algorithm application which produces a color for each vertex. In subject scheduling, it is assumed that the vertex is the subject and the teacher, while the edge is the class. In vertex coloring, graph vertices are colored so that there's no two neighboring vertices have the same color. The aim of this research was to arrange a lesson schedule so that problems do not occur such as clashes between teachers, subjects, and teaching hours. The method used in arranging this lesson schedule used the Welch-Powell Algorithm. The results obtained were using the Welch-Powell Algorithm to produce a lesson schedule every day where if there are two classes that have the same subject, they can meet the same day requirements but come in different hours and get a lesson schedule that has no clash between teachers, subjects, and teaching hours.

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

Cette thèse s'intéresse à l'égalité des preuves et des formules en logique linéaire, avec des contributions en particulier dans le fragment multiplicatif-additif de cette logique. En logique linéaire, et dans de nombreuses autres logiques (telle que la logique intuitionniste), on dispose de deux transformations sur les preuves : l'élimination des coupures et l'expansion des axiomes. On souhaite très souvent identifier deux preuves reliées par ces transformations, étant donné qu'elles le sont sémantiquement (dans un modèle catégorique par exemple). Cette situation est similaire à celle du λ-calcul où les termes sont identifiés à β-réduction et η-expansion près, opérations qui, par le prisme de la correspondance de Curry-Howard, se rapportent respectivement à l'élimination des coupures et à l'expansion des axiomes. Nous montrons ici que cette identification des preuves correspond exactement à l'identification des preuves à commutation de règle près, qui est une troisième opération sur les preuves bien connue et plus facile à manipuler. Notre démonstration considère uniquement la logique linéaire multiplicative-additive, même si nous conjecturons que ce résultat est vrai pour la logique linéaire dans son entièreté.Non seulement des preuves peuvent être identifiées à élimination des coupures et expansion des axiomes près, mais aussi des formules. Deux formules sont isomorphes si elles sont reliées par des preuves dont les compositions donnent l'identité, toujours à élimination des coupures et expansion des axiomes près. Ces formules sont alors réellement considérées comme les mêmes, et toute utilisation de l'une peut être remplacée par une utilisation de l'autre. Nous donnons une théorie équationnelle caractérisant exactement les formules isomorphes dans la logique linéaire multiplicative-additive. Un problème généralisant les isomorphismes est celui des rétractions, qui intuitivement correspondent aux couples de formules où la première peut être remplacée par la seconde - mais pas nécessairement la seconde par la première, contrairement aux isomorphismes. Étudier les rétractions est bien plus complexe, et nous avons caractérisé les rétractions des atomes dans le fragment multiplicatif de la logique linéaire.Pour l'étude des deux problèmes précédents, la syntaxe usuelle des preuves du calcul des séquents semble mal adaptée, car on considère des preuves à commutation de règle prés. Une partie de la logique linéaire possède une meilleure syntaxe dans ce cas : les réseaux de preuve, qui sont des graphes représentant des preuves quotientées par les commutations de règle. Cette syntaxe fut un outil indispensable pour caractériser isomorphismes et rétractions. Malheureusement, les réseaux de preuve ne sont pas (ou mal) définis en présence des unités. Pour nos problèmes, cette restriction conduit à une étude du cas sans unité dans les réseaux avec le coeur de la démonstration, précédé d'un travail en calcul des séquents pour prendre en compte les unités.Cette thèse développe par ailleurs une partie de la théorie des réseaux de preuve en fournissant une simple preuve du théorème de séquentialisation, qui relie les deux syntaxes des réseaux de preuve et du calcul des séquents, justifiant qu'elles décrivent les mêmes objets sous-jacents. Cette nouvelle démonstration s'obtient comme corollaire d'une généralisation du théorème de Yeo. Ce dernier résultat s'exprime entièrement dans la théorie des graphes aux arêtes colorées, et permet de déduire des preuves de sequentialisation pour différentes définitions de réseaux de preuve. Enfin, nous avons aussi formalisé les réseaux de preuve du fragment multiplicatif de la logique linéaire dans l'assistant de preuve Coq, avec en particulier une implémentation de notre nouvelle preuve de séquentialisation
This study is concerned with the equality of proofs and formulas in linear logic, with in particular contributions for the multiplicative-additive fragment of this logic. In linear logic, and as in many other logics (such as intuitionistic logic), there are two transformations on proofs: cut-elimination and axiom-expansion. One often wishes to identify two proofs related by these transformations, as it is the case semantically (in a categorical model for instance). This situation is similar to the one in the λ-calculus where terms are identified up to β-reduction and η-expansion, operations that, through the prism of the Curry-Howard correspondence, are related respectively to cut-elimination and axiom-expansion. We show here that this identification corresponds exactly to identifying proofs up to rule commutation, a third well-known operation on proofs which is easier to manipulate. We prove so only in multiplicative-additive linear logic, even if we conjecture such a result holds in full linear logic.Not only proofs but also formulas can be identified up to cut-elimination and axiom-expansion. Two formulas are isomorphic if there are proofs between them whose compositions yield identities, still up to cut-elimination and axiom-expansion. These formulas are then really considered to be the same, and every use of one can be replaced with one use of the other. We give an equational theory characterizing exactly isomorphic formulas in multiplicative-additive linear logic. A generalization of an isomorphism is a retraction, which intuitively corresponds to a couple of formulas where the first can be replaced by the second -- but not necessarily the other way around, contrary to an isomorphism. Studying retractions is more complicated, and we characterize retractions to an atom in the multiplicative fragment of linear logic.When studying the two previous problems, the usual syntax of proofs from sequent calculus seems ill-suited because we consider proofs up to rule commutation. Part of linear logic can be expressed in a better adapted syntax in this case: proof-nets, which are graphs representing proofs quotiented by rule commutation. This syntax was an instrumental tool for the characterization of isomorphisms and retractions. Unfortunately, proof-nets are not (or badly) defined with units. Concerning our issues, this restriction leads to a study of the unit-free case by means of proof-nets with the crux of the demonstration, preceded by a work in sequent calculus to handle the units. Besides, this thesis also develops part of the theory of proof-nets by providing a simple proof of the sequentialization theorem, which relates the two syntaxes of proof-net and sequent calculus, substantiating that they describe the same underlying objects. This new demonstration is obtained as a corollary of a generalization of Yeo's theorem. This last result is fully expressed in the theory of edge-colored graphs, and allows to recover proofs of sequentialization for various definitions of proof-nets. Finally, we also formalized proof-nets for the multiplicative fragment of linear logic in the proof assistant Coq, with notably an implementation of our new sequentialization proof

This thesis mainly focuses on algorithmic and combinatorial questions related to some geometric problems on graphs. In the last part of this thesis, a graph coloring problem is also discussed. Boxicity and Cubicity: These are graph parameters dealing with geomet-ric representations of graphs in higher dimensions. Both these parameters are known to be NP-Hard to compute in general and are even hard to approximate within an O(n1− ) factor for any > 0, under standard complexity theoretic assumptions. We studied algorithmic questions for these problems, for certain graph classes, to yield efficient algorithms or approximations. Our results include a polynomial time constant factor approximation algorithm for computing the cubicity of trees and a polynomial time constant (≤ 2.5) factor approximation algorithm for computing the boxicity of circular arc graphs. As far as we know, there were no constant factor approximation algorithms known previously, for computing boxicity or cubicity of any well known graph class for which the respective parameter value is unbounded. We also obtained parameterized approximation algorithms for boxicity with various edit distance parameters. An o(n) factor approximation algorithm for computing the boxicity and cubicity of general graphs also evolved as an interesting corollary of one of these parameterized algorithms. This seems to be the first sub-linear factor approximation algorithm known for computing the boxicity and cubicity of general graphs. Planar grid-drawings of outerplanar graphs: A graph is outerplanar, if it has a planar embedding with all its vertices lying on the outer face. We give an efficient algorithm to 2-vertex-connect any connected outerplanar graph G by adding more edges to it, in order to obtain a supergraph of G such that the resultant graph is still outerplanar and its pathwidth is within a constant times the pathwidth of G. This algorithm leads to a constant factor approximation algorithm for computing minimum height planar straight line grid-drawings of outerplanar graphs, extending the existing algorithm known for 2-vertex connected outerplanar graphs. n−1 3 Maximum matchings in triangle distance Delaunay graphs: Delau-nay graphs of point sets are well studied in Computational Geometry. Instead of the Euclidean metric, if the Delaunay graph is defined with respect to the convex distance function defined by an equilateral triangle, it is called a Trian-gle Distance Delaunay graph. TD-Delaunay graphs are known to be equivalent to geometric spanners called half-Θ6 graphs. It is known that classical Delaunay graphs of point sets always contain a near perfect matching, for non-degenerate point sets. We show that Triangle Distance Delaunay graphs of a set of n points in general position will always l m contain a matching of size and this bound is tight. We also show that Θ6 graphs, a class of supergraphs of half-Θ6 graphs, can have at most 5n − 11 edges, for point sets in general position. Heterochromatic Paths in Edge Colored Graphs: Conditions on the coloring to guarantee the existence of long heterochromatic paths in edge col-ored graphs is a well explored problem in literature. The objective here is to obtain a good lower bound for λ(G) - the length of a maximum heterochro-matic path in an edge-colored graph G, in terms of ϑ(G) - the minimum color degree of G under the given coloring. There are graph families for which λ(G) = ϑ(G) − 1 under certain colorings, and it is conjectured that ϑ(G) − 1 is a tight lower bound for λ(G). We show that if G has girth is at least 4 log2(ϑ(G))+2, then λ(G) ≥ ϑ(G)− 2. It is also proved that a weaker requirement that G just does not contain four-cycles is enough to guarantee that λ(G) is at least ϑ(G) −o(ϑ(G)). Other special cases considered include lower bounds for λ(G) in edge colored bipartite graphs, triangle-free graphs and graphs without heterochromatic triangles.

This thesis mainly focuses on algorithmic and combinatorial questions related to some geometric problems on graphs. In the last part of this thesis, a graph coloring problem is also discussed. Boxicity and Cubicity: These are graph parameters dealing with geomet-ric representations of graphs in higher dimensions. Both these parameters are known to be NP-Hard to compute in general and are even hard to approximate within an O(n1− ) factor for any > 0, under standard complexity theoretic assumptions. We studied algorithmic questions for these problems, for certain graph classes, to yield efficient algorithms or approximations. Our results include a polynomial time constant factor approximation algorithm for computing the cubicity of trees and a polynomial time constant (≤ 2.5) factor approximation algorithm for computing the boxicity of circular arc graphs. As far as we know, there were no constant factor approximation algorithms known previously, for computing boxicity or cubicity of any well known graph class for which the respective parameter value is unbounded. We also obtained parameterized approximation algorithms for boxicity with various edit distance parameters. An o(n) factor approximation algorithm for computing the boxicity and cubicity of general graphs also evolved as an interesting corollary of one of these parameterized algorithms. This seems to be the first sub-linear factor approximation algorithm known for computing the boxicity and cubicity of general graphs. Planar grid-drawings of outerplanar graphs: A graph is outerplanar, if it has a planar embedding with all its vertices lying on the outer face. We give an efficient algorithm to 2-vertex-connect any connected outerplanar graph G by adding more edges to it, in order to obtain a supergraph of G such that the resultant graph is still outerplanar and its pathwidth is within a constant times the pathwidth of G. This algorithm leads to a constant factor approximation algorithm for computing minimum height planar straight line grid-drawings of outerplanar graphs, extending the existing algorithm known for 2-vertex connected outerplanar graphs. n−1 3 Maximum matchings in triangle distance Delaunay graphs: Delau-nay graphs of point sets are well studied in Computational Geometry. Instead of the Euclidean metric, if the Delaunay graph is defined with respect to the convex distance function defined by an equilateral triangle, it is called a Trian-gle Distance Delaunay graph. TD-Delaunay graphs are known to be equivalent to geometric spanners called half-Θ6 graphs. It is known that classical Delaunay graphs of point sets always contain a near perfect matching, for non-degenerate point sets. We show that Triangle Distance Delaunay graphs of a set of n points in general position will always l m contain a matching of size and this bound is tight. We also show that Θ6 graphs, a class of supergraphs of half-Θ6 graphs, can have at most 5n − 11 edges, for point sets in general position. Heterochromatic Paths in Edge Colored Graphs: Conditions on the coloring to guarantee the existence of long heterochromatic paths in edge col-ored graphs is a well explored problem in literature. The objective here is to obtain a good lower bound for λ(G) - the length of a maximum heterochro-matic path in an edge-colored graph G, in terms of ϑ(G) - the minimum color degree of G under the given coloring. There are graph families for which λ(G) = ϑ(G) − 1 under certain colorings, and it is conjectured that ϑ(G) − 1 is a tight lower bound for λ(G). We show that if G has girth is at least 4 log2(ϑ(G))+2, then λ(G) ≥ ϑ(G)− 2. It is also proved that a weaker requirement that G just does not contain four-cycles is enough to guarantee that λ(G) is at least ϑ(G) −o(ϑ(G)). Other special cases considered include lower bounds for λ(G) in edge colored bipartite graphs, triangle-free graphs and graphs without heterochromatic triangles.

This chapter considers a new type of graph coloring known as edge coloring. It begins with a discussion of an idea by Scottish physicist Peter Guthrie Tait that led to edge coloring. Tait proved that the regions of every 3-regular bridgeless planar graph could be colored with four or fewer colors if and only if the edges of such a graph could be colored with three colors so that every two adjacent edges are colored differently. Tait thought that he had found a new way to solve the Four Color Problem. The chapter also examines the chromatic index of a graph, Vizing's Theorem, applications of edge colorings, and a class of numbers in graph theory called Ramsey numbers. Finally, it describes the Road Coloring Theorem which deals with traffic systems consisting only of one-way streets in which the same number of roads leave each location.

Determining the satisfiability of Boolean constraint-satisfaction problems with different types of constraints, that is hybrid constraints, is a well-studied problem with important applications. We study a new application of hybrid Boolean constraints, which arises in quantum computing. The problem relates to constrained perfect matching in edge-colored graphs. While general-purpose hybrid constraint solvers can be powerful, we show that direct encodings of the constrained-matching problem as hybrid constraints scale poorly and special techniques are still needed. We propose a novel encoding based on Tutte's Theorem in graph theory as well as optimization techniques. Empirical results demonstrate that our encoding, in suitable languages with advanced SAT solvers, scales significantly better than a number of competing approaches on constrained-matching benchmarks. Our study identifies the necessity of designing problem-specific encodings when applying powerful general-purpose constraint solvers.