Incident Edges Research Articles

The chromatic number of signed graphs with bounded maximum average degree

A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex v of a signed graph corresponds to changing the type of each edge incident to v.A homomorphism from a signed graph G to another signed graph H is a mapping φ:V(G)→V(H) such that, after switching some of the vertices of G, φ maps every edge of G to an edge of H of the same type. The chromatic number χs(G) of a signed graph G is the order of a smallest signed graph H such that there is a homomorphism from G to H.The maximum average degree mad(G) of a graph G is the maximum of the average degrees of all the subgraphs of G. We denote Mk the class of signed graphs with maximum average degree less than k and Pg the class of planar signed graphs of girth at least g.We prove: •χs(P7)≤5,•χs(M175)≤10 which implies χs(P5)≤10,•χs(M4−8q+3)≤q+1 with q a prime power congruent to 1 modulo 4.

On coloring of graphs of girth $2\l+1$ without longer odd holes

A hole is an induced cycle of length at least 4. Let $\l\ge~2$ be an integer and $r=\min\{\l-1,~\lfloor{l\over~2}\rfloor+1\}$, let ${\cal~G}_l$ denote the family of graphs which have girth $2\l+1$ and have no holes of odd length at least $2\l+3$, and let $G\in~{\cal~G}_{\l}$. For a vertex $u\in~V(G)$ and a nonempty set $S\subseteq~V(G)$, let $d(u,~S)=\min\{d(u,~v):v\in~S\}$, and let $L_i(S)=\{u\in~V(G)~\mbox{~and~}d(u,~S)=i\}$ for any integer $i\ge~0$. We show that if $G[S]$ is connected and $G[L_i(S)]$ is bipartite for each $i\in\{1,~\ldots,~r\}$, then $G[L_i(S)]$ is bipartite for each $i>0$, and consequently $\chi(G)\le~4$, where $G[S]$ denotes the subgraph induced by $S$.Let $\theta^-$ be the graph obtained from the Petersen graph by deleting three vertices which induce a path, let $\theta^+$ be the graph obtained from the Petersen graph by deleting two adjacent vertices, and let $\theta$ be the graph obtained from $\theta^+$ by removing an edge incident with two vertices of degree 3. For a non-Petersen graph $G\in{\cal~G}_2$, we show that if $G$ is 3-connected and has no unstable 3-cutset then $G$ must induce either $\theta$ or $\theta^-$ but does not induce $\theta^+$. As corollaries, $\chi(G)\le~3$ for every graph $G$ of ${\cal~G}_2$ that induces neither $\theta$ nor $\theta^-$, and minimal non-3-colorable graphs of ${\cal~G}_2$ induce no $\theta^+$.

Complete Graph Drawings up to Triangle Mutations

The main result of the paper can be stated in the following way: a complete graph drawing in the sphere, where two edges have at most one common point, which is either a crossing or a common endpoint, and no three edges share a crossing, is determined by the circular ordering of edges at each vertex, or equivalently by the set of pairs of edges that cross, up to homeomorphism and a sequence of triangle mutations. A triangle mutation (or switch, or flip) consists in passing an edge across the intersection of two other edges, assuming the three edges cross each other and the region delimited by the three edges has an empty intersection with the drawing. Equivalently, the result holds for a drawing in the plane assuming the drawing is given with a pair of edges indicating where the unbounded region is. The proof is constructive, based on an inductive algorithm that adds vertices and their incident edges one by one (actually yielding an extra property for the sequence of triangle mutations). This result generalizes Ringel’s theorem on uniform pseudoline arrangements (or rank 3 uniform oriented matroids) to complete graph drawings. We also apply this result to plane projections (or visualizations) of a geometric spatial complete graph, in terms of the rank 4 uniform oriented matroid defined by its vertices. Independently, we prove that, for a complete graph drawing, the set of pairs of edges that cross determine, by first order logic formulas, the circular ordering of edges at each vertex, as well as further information.

The adjacent vertex distinguishing edge choosability of planar graphs with maximum degree at least 11

An adjacent vertex distinguishing edge coloring is a proper edge coloring such that any two adjacent vertices have distinct sets consisting of colors of their incident edges. It was conjectured that if G is a connected graph with at least 6 vertices, then χa′(G)≤Δ(G)+2. In this paper, we consider the list version of adjacent vertex distinguishing edge coloring and show that if G is a planar graph without isolated edges, then cha′(G)≤max{13,Δ(G)+2}, where cha′(G) is the adjacent vertex distinguishing edge choosability. This result implies that the above conjecture holds for planar graphs with maximum degree at least 11. Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

Total Coloring and Total Matching: Polyhedra and Facets

A total coloring of a graph G=(V,E) is an assignment of colors to vertices and edges such that neither two adjacent vertices nor two incident edges get the same color, and, for each edge, the end-points and the edge itself receive different colors. Any valid total coloring induces a partition of the elements of G into total matchings, which are defined as subsets of vertices and edges that can take the same color. In this paper, we propose Integer Linear Programming models for both the Total Coloring and the Total Matching problems, and we study the strength of the corresponding Linear Programming relaxations. The total coloring is formulated as the problem of finding the minimum number of total matchings that cover all the graph elements. This covering formulation can be solved by a Column Generation algorithm, where the pricing subproblem corresponds to the Weighted Total Matching Problem. Hence, we study the Total Matching Polytope. We introduce three families of nontrivial valid inequalities: vertex-clique inequalities based on standard clique inequalities of the Stable Set Polytope, congruent-2k3 cycle inequalities based on the parity of the vertex set induced by the cycle, and even-clique inequalities induced by complete subgraphs of even order. We prove that congruent-2k3 cycle inequalities are facet-defining only when k=4, while the vertex-clique and even-cliques are always facet-defining. Finally, we present preliminary computational results of a Column Generation algorithm for the Total Coloring Problem and a Cutting Plane algorithm for the Total Matching Problem.

The Density of Fan-Planar Graphs

A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by three forbidden patterns, one of which is the configuration where $e$ is crossed by two independent edges and the other two where $e$ is crossed by two incident edges in a way that encloses some endpoint of $e$. A topological drawing is simple if any two edges have at most one point in common.
 Fan-planar graphs are a new member in the ever-growing list of topological graphs defined by forbidden intersection patterns, such as planar graphs and their generalizations, Turán-graphs and Conway's thrackle conjecture. Hence fan-planar graphs fall into an important field in combinatorial geometry with applications in various areas of discrete mathematics.
 As every $1$-planar graph is fan-planar and every fan-planar graph is $3$-quasiplanar, they also fit perfectly in a recent series of works on nearly-planar graphs from the areas of graph drawing and combinatorial embeddings.
 In this paper we show that every fan-planar graph on $n$ vertices has at most $5n-10$ edges, even though a fan-planar drawing may have a quadratic number of crossings. Our bound, which is tight for every $n \geq 20$, indicates how nicely fan-planar graphs fit in the row with planar graphs ($3n-6$ edges) and $1$-planar graphs ($4n-8$ edges). With this, fan-planar graphs form an inclusion-wise largest non-trivial class of topological graphs defined by forbidden patterns, for which the maximum number of edges on $n$ vertices is known exactly.
 We demonstrate that maximum fan-planar graphs carry a rich structure, which makes this class attractive for many algorithms commonly used in graph drawing. Finally, we discuss possible extensions and generalizations of these new concepts.

Total outer connected vertex-edge domination

A vertex [Formula: see text] of a graph is said to [Formula: see text]-[Formula: see text] dominate every edge incident with [Formula: see text], as well as every edge incident to vertices adjacent to [Formula: see text]. A subset [Formula: see text] is a total outer connected vertex-edge dominating set of a graph [Formula: see text] if every edge in [Formula: see text] is [Formula: see text]-[Formula: see text] dominated by a vertex in [Formula: see text], the subgraph induced by [Formula: see text] has no isolated vertices and the subgraph induced by [Formula: see text] is connected. We initiate the study of total outer connected vertex-edge domination in graphs. We show that the decision problem for total outer-connected vertex-edge domination problem is [Formula: see text]-Complete even for bipartite graphs. We prove that for every tree of order [Formula: see text] with [Formula: see text] leaves, [Formula: see text] and characterize the trees attaining the lower bound. We also study the effect of edge removal, edge addition and edge subdivision on total outer connected vertex-edge domination number of a graph.

Polynomial-time Algorithm for Maximum Weight Independent Set on P 6 -free Graphs

In the classic Maximum Weight Independent Set problem, we are given a graph G with a nonnegative weight function on its vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P 6 -free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P 5 -free graphs of Lokshtanov et al. [ 15 ] and the quasipolynomial-time algorithm on P 6 -free graphs of Lokshtanov et al. [ 14 ]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: For every maximal independent set I in the graph, ℱ contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I .

A sharp upper bound on the spectral radius of C5-free /C6-free graphs with given size

Let Sn,2 be the graph obtained by joining each vertex of K2 to n−2 isolated vertices, and let Sn,2− be the graph obtained from Sn,2 by deleting an edge incident to a vertex of degree two. Recently, Zhai, Lin and Shu [20] showed that ρ(G)≤1+4m−32 for any C5-free graph of size m≥8 or C6-free graph of size m≥22, with equality if and only if G≅Sm+32,2 (possibly, with some isolated vertices). However, this bound is sharp only for odd m. Motivated by this, we want to obtain a sharp upper bound of ρ(G) for C5-free or C6-free graphs with m edges. In this paper, we prove that if G is a C5-free graph of even size m≥14 or C6-free graph of even size m≥74, and G contains no isolated vertices, then ρ(G)≤ρ˜(m), with equality if and only if G≅Sm+42,2−, where ρ˜(m) is the largest root of x4−mx2−(m−2)x+(m2−1)=0.

On Rainbow Cycles and Proper Edge Colorings of Generalized Polygons

An edge coloring of a simple graph G is said to be proper rainbow-cycle-forbidding (PRCF, for short) if no two incident edges receive the same color and for any cycle in G, at least two edges of that cycle receive the same color. A graph G is defined to be PRCF-good if it admits a PRCF edge coloring, and G is deemed PRCF-bad otherwise. In recent work, Hoffman, et al. study PRCF edge colorings and find many examples of PRCF-bad graphs having girth less than or equal to 4. They then ask whether such graphs exist having girth greater than 4. In our work, we give a straightforward counting argument showing that the Hoffman-Singleton graph answers this question in the affirmative for the case of girth 5. It is then shown that certain generalized polygons, constructed of sufficiently large order, are also PRCF-bad, thus proving the existence of PRCF-bad graphs of girth 6, 8, 12, and 16.

Оценка эффективности метода синтеза мультисервисной сети связи

The article proposes directions for evaluating the efficiency of the multiservice communication network structure variant, and on the basis of euclidean heuristics, an approach to evaluating the efficiency of the method of synthesizing a multiservice packet-switched communication network consisting in searching for a structure on a limited set of vertices connected by minimum edges by weight, which is a set of independent spanning trees on a limited subset of minimum connections. It is proposed to assess the effectiveness of the method of synthesizing a multiservice communication network for connected saturated graphs, since an accurate analysis of the effectiveness of euclidean heuristics is a difficult analytical task and very significantly depends on both the number and location of vertices and the number of edges incident by it. The given version of estimation of efficiency of method of synthesis of multiservice communication network is based on results of multivariate computational experiment for random saturated graphs, upper limits of characteristics of which are obtained from normative and technical documentation by evaluation analysis of values of parameters of promising multiservice communication networks. Assessment of the set of main and additional actions when searching for the required number of independent spanning trees, the structure of the multiservice communication network in comparison with the Prima algorithm allows us to conclude that, that the synthesis method under consideration enables to obtain a rational solution to the structure synthesis problem, constituting a multiservice communication network in an acceptable time compared to known synthesis methods; which, according to the results of statistical tests for the considered practically important subset of implementation options for promising multiservice communication networks, is two to three times less than the estimate of the time for performing a search on a set of all vertices.

Modular Irregular Labeling on Double-Star and Friendship Graphs

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

Anti-magic like labelling of marigold graph

In this paper, we present a new family of graphs called Marigold graphs and introduce a new labelling method similar to the anti-magic labelling. The Marigold graph is generated from any number of copies of fully binary trees which are going through concentric circles. All copies of trees are connected to a middle vertex and the height of the Marigold graph is increasing with <em>n</em> concentric circles. One copy is considered as one petal in the marigold graph. A Marigold graph with copies (petals) and height (number of concentric circles) <em>k</em> is <em>M ⁿ_k</em> denoted by . The labelling method is defined as follows: A graph with ‘<em>m</em>’ edges and ‘<em>n</em>’ vertices is labelled as an injection from the set of edges to the integers {1, …, <em>x</em>} such that all ‘<em>n</em>’ vertex sums are pairwise distinct, where the vertex sum is the sum of labels of all edges incident with that vertex. In our work, for edge labelling, we consider the petals one by one and denote the <em>r</em>^th edge at <em>k</em>^th level as <em>e^k_r</em> , and define a function to label edges of the first petal. Then define the new labelling method for other petals, such that for <em>n</em>^th petal, edge labelling is starting with <em>J_n</em>_–1+ 1 (where <em>J_n</em>_–1is the summation of all edge values in (<em>n</em> –1)^th petal, ∑<em>ⁱ</em>⁼¹<em>_m</em> (<em>n</em>–1, <em>i</em>) = <em>J_n</em>_–1and continue the labelling as a monotonically increasing sequence. We discuss some illustrative examples that might be used for studying the Anti-magic like labelling of Marigold graphs.

A tight bound for shortest augmenting paths on trees

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree T=(W⊎B,E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. It was conjectured by Chaudhuri et al. [6] that the total length of all shortest augmenting paths found is O(nlog⁡n). In this paper we prove a tight O(nlog⁡n) upper bound for the total length of shortest augmenting paths for trees improving over O(nlog2⁡n) bound [3].

Small signal stability analysis of VSC based DC systems using graph theory

This paper presents a new method to evaluate the small signal stability based on state space and graph theory, which not only reduces the computational burden of high-dimensional system, but also has the ability to cope with the uncertain parameters and system structure. The system is first represented by a directed graph. Each node in the directed graph denotes a multi-input multi-output subsystem represented by state, input, and output matrices, and constrained by norm bounded uncertainties. The subsystems are connected by edges through node-edge incidence matrix and edge connection matrices. Then we propose the small signal stability conditions associated with the certain node dynamics, edge connection matrices, the norm bounds of uncertainty, and the maximum degree of the system, independent of the system structure and the number of nodes and edges. We apply our methods to the VSC based DC systems with uncertain parameters and structure. The comparative analysis of simulations results of Bode diagram, state matrix eigenvalues, and the new method validates the effectiveness of the proposed stability conditions.

Game Self-organization of Hamiltonian Cycle of the Graph

This paper proposes a new application of the stochastic game model to solve the problem of self- organization of the Hamiltonian cycle of a graph. To do this, at the vertices of the undirected graph are placed game agents, whose pure strategies are options for choosing one of the incident edges. A random selection of strategies by all agents forms a set of local paths that begin at each vertex of the graph. Current player payments are defined as loss functions that depend on the strategies of neighboring players that control adjacent vertices of the graph. These functions are formed from a penalty for the choice of opposing strategies by neighboring players and a penalty for strategies that have reduced the length of the local path. Random selection of players’ pure strategies is aimed at minimizing their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution built on the basis of dynamic vectors of mixed strategies. The elements of the vectors of mixed strategies are the probabilities of choosing the appropriate pure strategies that adaptively take into account the values of current losses. The formation of vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which the gradient method of stochastic approximation is used. During the game, the method increases the value of the probabilities of choosing those pure strategies that lead to a decrease in the functions of average losses. For given methods of forming current payments, the result of the stochastic game is the formation of patterns of self-organization in the form of cyclically oriented strategies of game agents. The conditions of convergence of the recurrent method to collectively optimal solutions are ensured by observance of the fundamental conditions of stochastic approximation. The game task is extended to random graphs. To do this, the vertices are assigned the probabilities of recovery failures, which cause a change in the structure of the graph at each step of the game. Realizations of a random graph are adaptively taken into account when searching for Hamiltonian cycles. Increasing the probability of failure slows down the convergence of the stochastic game. Computer simulation of the stochastic game provided patterns of self-organization of agents’ strategies in the form of several local cycles or a global Hamiltonian cycle of the graph, depending on the ways of forming the current losses of players. The reliability of experimental studies is confirmed by the repetition of implementations of self-organization patterns for different sequences of random variables. The results of the study can be used in practice for game-solving NP-complex problems, transport and communication problems, for building authentication protocols in distributed information systems, for collective decision-making in conditions of uncertainty.

On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion

All graph in this paper are simple, finite, and connected. Let be a labeling of a graph . The function is called antimagic rainbow edge labeling if for any two vertices and , all internal vertices in path have different weight, where the weight of vertex is the sum of its incident edges label. The vertex weight denoted by for every . If G has a antimagic rainbow edge labeling, then is a antimagic rainbow vertex connection, where the every vertex is assigned with the color . The antimagic rainbow vertex connection number of , denoted by , is the minimum colors taken over all rainbow vertex connection induced by antimagic rainbow edge labeling of . In this paper, we determined the exact value of the antimagic rainbow vertex connection number of path ( ), wheel ( ), friendship ( ), and fan ( ).

Angle Covers: Algorithms and Complexity

Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering- an angle- such that each edge participates in at least one such pair. We show that any graph of maximum degree 4 admits an angle cover, give a poly-time algorithm for deciding if a graph without a degree-3 vertex has an angle cover, and prove that, given a graph of maximum degree 5, it is NP-hard to decide whether it admits an angle cover. We also consider extensions of the angle cover problem where every vertex selects a fixed number $a&gt;1$ of angles or where an angle consists of more than two consecutive edges. We show an application of angle covers to the problem of deciding if the 2-blowup of a planar graph has isomorphic thickness 2.

Product Antimagic Labeling of Caterpillars

Let G be a graph with m edges. A product antimagic labeling of G is a bijection from the edge set E G to the set 1,2 , … , m such that the vertex-products are pairwise distinct, where the vertex-product of a vertex v is the product of labels on the incident edges of v . A graph is called product antimagic if it admits a product antimagic labeling. In this paper, we will show that caterpillars with at least three edges are product antimagic by an O m log m algorithm.

Resin Bonded Diamond grinding wheels conditioning using SiC rotary dresser

Superabrasive grinding of advanced materials parts with complex geometry is a critical issue for aerospace or energy industry. Likewise, conditioning of superabrasive is a key process in order to obtain the most effective grinding. While from industrial point of view the conditioning parameters and dressers are designed from the experience, from scientific point of view the last works are focused on achieving guidelines for optimising conditioning process. In this sense, the present work is focussed on the conditioning process of diamond resin bonded grinding wheels. To this end, SiC rotary dressers of different hardness and applying down-cut and up-cut are tested in order to achieve the best conditions. On the one hand, the wheel surface is deeply analyse. A software is developed in order to analyse the grain pull out, the new abrasive grains and the size evolution and grain concentration during conditioning. On the other hand, the wheel geometry is also measured due to the importance of geometry loss during the conditioning process on profile grinding wheels. The results shown that down-cut conditions are more aggressive obtaining a better wheel surface recovery and the wheel geometry of the incidence edge is lost, requiring a subsequent truing process to obtain the wheel profile. Finally, it is noteworthy that the methodology developed for the analysis of the diamond resin bonded grinding wheels is suitable for all types of grinding wheels even rotary dressers.