Incident Edges Research Articles

A Note of Reliability Analysis of SM-λ in Folded-Crossed Hypercube with Conditional Faults

The fault tolerance of an interconnection network of parallel and distributed systems can be evaluated by various topological parameters of its underlying graph [Formula: see text], with strong Menger edge connectivity being a vital parameter in this regard. A connected graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text]) if it connects any pair of vertices [Formula: see text] and [Formula: see text] with [Formula: see text] number of edge-disjoint paths. Under the uniform distribution of faults in a large interconnection network, it is improbable that each faulty edge incident to a vertex will occur simultaneously. Thus, [Formula: see text]-strongly Menger edge connected of order [Formula: see text] was introduced in 2018 by He et al. Here, [Formula: see text] is called as [Formula: see text]-strongly Menger edge connected of order [Formula: see text], if [Formula: see text] remains SM-[Formula: see text], where [Formula: see text] is an arbitrary edge set in a graph [Formula: see text] with [Formula: see text] and the minimum degree of the remaining graph [Formula: see text]. The largest [Formula: see text] keeping the property of [Formula: see text] being SM-[Formula: see text] is denoted as [Formula: see text]. Among variants of hypercube, the [Formula: see text]-dimensional folded-crossed hypercube [Formula: see text] attracts attention in recent years. In this paper, we focus on calculating the exact value of the maximum conditional edge-fault-tolerant number of order [Formula: see text] of [Formula: see text], [Formula: see text] for two integers [Formula: see text] and [Formula: see text].

Complemented zero-divisor graphs associated with finite commutative semigroups

Let S be a commutative semigroup with zero. The zero-divisor graph associated with S is the simple graph whose vertex set is given by the nonzero zero-divisors of S where two distinct vertices are adjacent if and only if their product is zero. A graph is called complemented if every vertex has an incident edge which is not the edge of any three cycle. Complemented zero-divisor graphs which are associated with a finite commutative semigroup are described in terms of their clique number. The algebraic properties of the finite semigroups S whose zero-divisor graphs are complemented are described in terms of the clique number of the graph. Infinite families of complemented zero-divisor graphs associated with a finite commutative semigroup are given.

Robust circuitry-based scores of structural importance of human brain areas.

We consider the 1015-vertex human consensus connectome computed from the diffusion MRI data of 1064 subjects. We define seven different orders on these 1015 graph vertices, where the orders depend on parameters derived from the brain circuitry, that is, from the properties of the edges (or connections) incident to the vertices ordered. We order the vertices according to their degree, the sum, the maximum, and the average of the fiber counts on the incident edges, and the sum, the maximum and the average length of the fibers in the incident edges. We analyze the similarities of these seven orders by the Spearman correlation coefficient and by their inversion numbers and have found that all of these seven orders have great similarities. In other words, if we interpret the orders as scoring of the importance of the vertices in the consensus connectome, then the scores of the vertices will be similar in all seven orderings. That is, important vertices of the human connectome typically have many neighbors connected with long and thick axonal fibers (where thickness is measured by fiber numbers), and their incident edges have high maximum and average values of length and fiber-number parameters, too. Therefore, these parameters may yield robust ways of deciding which vertices are more important in the anatomy of our brain circuitry than the others.

Archimedean toroidal maps and their edge covers

Abstract The automorphism group of a map on a surface acts naturally on its flags (triples of incident vertices, edges, and faces). We will study the action of the automorphism group of a map on its edges. A map is semi-equivelar if all of its vertices have the same type of face-cycles. A semi-equivelar toroidal map refers to a semi-equivelar map embedded on a torus. If a map has k edge orbits under its automorphism group, it is referred to as a k-edge orbital or k-orbital. Specifically, it is referred to as an edge-transitive map if k = 1 {k=1} . If any two edges have the same edge-symbol, a map is said to be edge-homogeneous. Every edge-homogeneous toroidal map has an edge-transitive cover, as proved in [A. Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars Math. Contemp. 4 2011, 2, 385–402]. In this article, we show the existence and give a classification of k-edge covers of semi-equivelar toroidal maps.

Thrackles, Superthrackles and the Hanani-Tutte Theorem

A thrackle is a drawing of a graph in which any two vertex-disjoint edges cross exactly once and incident edges do not cross. A graph that has a thrackle drawing is thracklable. In the past three decades, mathematicians investigated a number of variations of thrackles by relaxing or changing the required number of crossings on edges or restricting the placement of vertices on the surface (e.g., restricting to outerdrawings, in which vertices are restricted to a boundary curve of the surface). A graph is superthracklable if it can be drawn so that any two edges cross exactly once. We provide a simple proof for the following: a graph can be drawn on the plane so that every two edges cross an odd number of times if and only if it is superthracklable on the plane. A graph is outersuperthracklable if it has a drawing on a disc in which every vertex is on the boundary of the disc and every pair of edges cross exactly once. We introduce some natural generalisations of outersuperthracklable graphs and we show that these classes of graphs are equivalent. Lastly, using the Hanani-Tutte characterisation of planar graphs we show that for any surface $\Sigma$, there is a relationship between the class of graphs that are not embeddable on $\Sigma$ and the class of graphs that are not superthracklable with respect to $\Sigma$. More specifically, we show how to construct, from any forbidden minor $G$ for embeddability in $\Sigma$, two infinite families of graphs that are not superthracklable with respect to $\Sigma$. This sheds further light on the relationship between some of Archdeacon and Stor's forbidden configurations for superthrackles and Kuratowski's forbidden minors for planarity.

Algorithmic Aspects of Vertex-edge Domination in Some Graphs

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V(G)\) ve-dominates every edge \(uv\) incident to \(v\), as well as every edge adjacent to these incident edges. A set \(D\subseteq V(G)\) is a vertex-edge dominating set if every edge of \(E(G)\) is ve-dominated by a vertex of \(D.\) The MINIMUM VERTEX-EDGE DOMINATION problem is to find a vertex-edge dominating set of minimum cardinality. A linear time algorithm to find the minimum vertex-edge dominating set for proper interval graphs is proposed. The vertex-edge domination problem is proved to be APX-complete for bounded-free graphs and NP-Complete for Chordal bipartite and Undirected Path graphs.

Complete bipartite graphs without small rainbow subgraphs

Motivated by bipartite Gallai–Ramsey type problems, we consider edge-colorings of complete bipartite graphs without rainbow tree and matching. Given two graphs G and H, and a positive integer k, define the bipartite Gallai–Ramsey numberbgrk(G:H) as the minimum number of vertices n such that n2≥k and for every N≥n, any coloring (using all k colors) of the complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first describe the structures of a complete bipartite graph Kn,n without rainbow P4+ and 3K2, respectively, where P4+ is the graph consisting of a P4 with one extra edge incident with an interior vertex. Furthermore, we determine the exact values or upper and lower bounds on bgrk(G:H) when G is a 3-matching or a 4-path or P4+, and H is a bipartite graph.

H-colorings for 4-regular graphs

For a vertex a in H, we denote by ∂H(a) the set of edges incident with a. An H-coloring of a graph G is a proper edge-coloring f:E(G)→E(H) such that for each vertex u in G, there exists a vertex a in H such that f(∂G(u))=∂H(a). In this paper, we introduce some graphs H such that H-colorability of 4-regular graphs is meaningful and related to some of their properties, such as an even cycle decomposition of size 3 and an even 2-factor. In addition, we formulate a conjecture on H-coloring of the line graph of some cubic graphs. Partial solutions to the conjecture are also presented.

Upper bound on saturation time of metric graphs by intervals moving on them

In this paper we study a dynamical system of intervals moving on a metric graph with incommensurable edge lengths. This system can be generally viewed as a collection of congruent intervals moving around the network at unit speed, propagating along all respective incident edges whenever they pass through a vertex. In particular, we analyse the phenomenon of saturation: a state when the entire graph is covered by the moving intervals. Our main contributions are as follows: (1) we prove the existence of the finite moment of permanent saturation for any metric graph with incommensurable edge lengths and any positive length of the intervals; (2) we present an upper bound on the moment of permanent saturation. To show the validity of our results, we reduce the system of moving intervals to the system of dispersing moving points, for the analysis of which we mainly use methods from discrepancy theory and number theory, in particular Kronecker sequences and the famous Three Gap Theorem.

Algorithmic Aspects of Total Vertex-Edge Domination in Graphs

A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.

Group-supermagic labeling of Cartesian products of two even cycles

A Γ-supermagic labeling of a graph G=(V,E) with |E|=q is a bijection from E to an Abelian group Γ of order q such that the sum of labels of all incident edges of every vertex x∈V is equal to the same element μ∈Γ. An existence of a Γ-supermagic labeling of Cartesian product of two cycles Cm□Cn for every m,n≥3 by Z2mn and for any two odd cycles Cm□Cn for any m,n≥3 and any Abelian group Γ of order 2mn was proved recently. In this paper we completely solve the case of m,n both even and present a labeling method for all Abelian groups of order 2mn, where m,n are even and greater than two.

On gap-labellings of some families of graphs

This text addresses the proper gap-labelling problem in both its edge and vertex variants. Consider an assignment of integer values {1,…,k} to either the edges or vertices of a graph. In the edge-labelling variant, a vertex with degree at least two has its colour defined by the largest difference, or the largest “gap”, among the labels of its incident edges; vertices of degree 1 receive as induced colour the label assigned to its (only) incident edge. For the vertex variant, the gaps are defined by the largest difference among the labels in the vertices neighbourhood, and degree-one vertices receive as colour the label assigned to their (only) neighbour. A gap-labelling is proper if the obtained colouring is proper and the least number k of labels for which a graph admits a proper gap-labelling of its edges (vertices) is called the edge-gap (vertex-gap) number of the graph. In this article, we study both parameters, from a structural and algorithmic point of view, for traditional families of graphs such as Cycles, Crowns and Wheels. We also consider Unicyclic graphs and several families of Snarks.

Graphs with unique minimum vertex-edge dominating sets

A vertex u of a graph G = (V, E), ve-dominates every edge incident to u, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or a ved-set for short) if every edge of E is ve-dominated by at least one vertex of S. The vertex-edge domination number is the minimum cardinality of a ved-set in G. In this paper, we investigate the graphs having unique minimum ved-sets that we will call UVED-graphs. We start by giving some basic properties of UVED-graphs. For the class of trees, we establish two equivalent conditions characterizing UVED-trees which we subsequently complete by providing a constructive characterization.

The chromatic number of 2-edge-colored and signed graphs of bounded maximum degree

A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. A homomorphism from a 2-edge-colored graph G to a 2-edge-colored graph H is a mapping φ:V(G)→V(H) that maps every edge in G to an edge of the same type in H. Switching a vertex v of a 2-edge-colored or signed graph corresponds to changing the type of each edge incident to v. There is a homomorphism from the signed graph G to the signed graph H if after switching some subset of the vertices of G there is a 2-edge-colored homomorphism from G to H.The chromatic number of a 2-edge-colored (resp. signed) graph G is the order of a smallest 2-edge-colored (resp. signed) graph H such that there is a homomorphism from G to H. The chromatic number of a class of graphs is the maximum of the chromatic numbers of the graphs in the class.We study the chromatic numbers of 2-edge-colored and signed graphs (connected and not necessarily connected) of a given bounded maximum degree. More precisely, we provide exact bounds for graphs with a maximum degree 2. We then propose specific lower and upper bounds for graphs with a maximum degree 3, 4, and 5. We finally propose general bounds for graphs of maximum degree k, for every k.

Atomistic evaluation of the dislocation transmission across tilt and twist low-angle grain boundaries in body-centered cubic iron

This study systematically and qualitatively evaluates the resistance for dislocation transmission across tilt/twist low-angle grain boundaries (LAGBs), which mainly consist of edge/screw dislocations (“GB dislocations”). The motion of the incident lattice edge dislocation is disturbed by the LAGBs, while the edge dislocation is transmitted to the adjacent grain by applying large shear stress if the tilt and twist angles are sufficiently low. It was observed that the twist LAGB is a stronger obstacle compared to the tilt LAGB for the incident lattice edge segment even though the misorientation angle of both tilt and twist LAGBs is the same because of the reconstruction of the GB-dislocation structures at the twist LAGB owing to the incident lattice dislocation. When we compare the incident lattice edge and screw components, the barrier effect of the GB on the transmission is stronger for the screw component than for the edge component. The stronger barrier effect can be explained mainly by the higher line tension of the screw component based on the dislocation bow-out model. Accordingly, the dominant factors of the GB-strengthening effect in LAGB are the incident lattice dislocation type, the structure of GB dislocations, and dislocation–dislocation reactions. These results indicate that the GB-strengthening factors are dominated by the type of incident lattice dislocation and the GB-dislocation character in addition to the macroscopic misorientation of the GB.

Local edge (a, d) –antimagic coloring on sunflower, umbrella graph and its application

Suppose a graph G = (V, E) is a simple, connected and finite graph with vertex set V(G) and an edge set E(G). The local edge antimagic coloring is a combination of local antimagic labelling and edge coloring. A mapping f∶ V (G)→ {1, 2, ..., |V (G)|} is called local edge antimagic coloring if every two incident edges e_1and e_2, then the edge weights of e_1and e_2 maynot be the same, w(e_1) ≠ w(e_2), with e = uv ∈ G, w(e) = f(u)+ f(v) with the rule that the edges e are colored according to their weights, w_e. Local edge antimagic coloring has developed into local (a,d)-antimagic coloring. Local antimagic coloring is called local (a,d)-antimagic coloring if the set of edge weights forms an arithmetic sequence with a as an initial value and d as a difference value. The graphs used in this study are sunflower graphs and umbrella graphs. This research will also discuss one of the applications of local edge (a,d)-antimagic coloring, namely the design of the Sidoarjo line batik motif. The result show that χ_le(3,1) (Sf_n) = 3n and χ_le(3n/2,1) (U_(m,n) ) = m+1 . The local (a,d)-antimagic coloring is formed into a batik motif design with characteristics from the Sidoarjo region.

Tight description of faces of triangulations on the torus

The degree d(x) of a vertex or face x in a graph G is the number of incident edges. A face f=v1…vd(f) in a graph G on the plane or other orientable surface is of type (k1,k2,…), where k1≤k2≤…, if d(vi)≤ki for each i. By δ we denote the minimum vertex-degree of G.In 1989, Borodin confirmed Kotzig's conjecture of 1963 that every plane graph with minimum degree δ equal to 5 has a (5,5,7)-face or a (5,6,6)-face, where all parameters are tight. Recently, we proved that every torus triangulation with δ≥5 has a face of one of the types (5,5,8), (5,6,7), or (6,6,6), which is tight.It follows from the classical theorem by Lebesgue (1940) that every plane triangulation with δ≥4 has a 3-face of types (4,4,∞), (4,5,19), (4,6,11), (4,7,9), (5,5,9), or (5,6,7).In 1999, Jendrol' gave a similar description: “(4,4,∞), (4,5,13), (4,6,17), (4,7,8), (5,5,7), (5,6,6)” and conjectured that “(4,4,∞), (4,5,10), (4,6,15), (4,7,7), (5,5,7), (5,6,6)” holds.In 2002, Lebesgue's description was strengthened by Borodin to “(4,4,∞), (4,5,17), (4,6,11), (4,7,8), (5,5,8), (5,6,6)”.In 2014, Borodin and Ivanova obtained the following tight description, which, in particular, disproves the above mentioned conjecture by Jendrol': “(4,4,∞), (4,5,11), (4,6,10), (4,7,7), (5,5,7), (5,6,6)”, and recently proved another tight description of faces in plane triangulations with δ≥4: “(4,4,∞), (4,6,10), (4,7,7), (5,5,8), (5,6,7)”.It follows from Lebesgue's theorem of 1940 that every plane quadrangulation with δ≥3 has a face of one of the types (3,3,3,∞), (3,3,4,11), (3,3,5,7), (3,4,4,5). Recently, Borodin and Ivanova improved this description to “(3,3,3,∞), (3,3,4,9), (3,3,5,6), (3,4,4,5)”, where all parameters except possibly 9 are best possible and 9 cannot go down below 8.In 1995, Avgustinovich and Borodin proved the following tight description of the faces of torus quadrangulations with δ≥3: “(3,3,3,∞), (3,3,4,10), (3,3,5,7), (3,3,6,6), (3,4,4,6), (4,4,4,4)”, which also holds for each higher surface provided that its quadrangulation is large enough.Recently, Borodin and Ivanova proved that every triangulation with δ≥4 of the torus has a face of one of the types (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), which description is tight.The purpose of this paper is to prove that every triangulation with δ≥3 on the torus has a face of one of the types (3,6,24), (3,8,16), (3,12,12), (4,4,∞), (4,6,12), (4,8,8), (5,5,8), (5,6,7), or (6,6,6), where all parameters are best possible.

Splitting Vertices in 2-Layer Graph Drawings.

Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines (layers), and their relationships (edges) are represented by segments connecting vertices. Methods for constructing 2-layer drawings often try to minimize the number of edge crossings. We use vertex splitting to reduce the number of crossings, by replacing selected vertices on one layer by two (or more) copies and suitably distributing their incident edges among these copies. We study several optimization problems related to vertex splitting, either minimizing the number of crossings or removing all crossings with fewest splits. While we prove that some variants are ${\mathsf {NP}}$NP-complete, we obtain polynomial-time algorithms for others. We run our algorithms on a benchmark set of bipartite graphs representing the relationships between human anatomical structures and cell types.

IC-planar graphs are odd-10-colorable

An odd-k-coloring of a graph G is a proper k-coloring such that for every vertex v∈V(G) with d(v)≥1, there exists a color occurring odd times in its neighbors. The odd chromatic number, denote by χo(G), of G is the smallest integer k so that G admits an odd-k-coloring. Call a graph G IC-planar if it can be drawn in the plane so that every edge is crossed by at most once and every vertex has at most one incident edge being crossed. It was shown that every planar graph G has χo(G)≤8. In this paper we show that every IC-planar graph G satisfies χo(G)≤10.

Total distance vertex irregularity strength of some corona product graphs

A distance vertex irregular total k -labeling of a simple undirected graph G = G ( V , E ) , is a function f : V ( G )∪ E ( G )→{1, 2, …, k } such that for every pair vertices u , v ∈ V ( G ) and u ≠ v , the weights of u and v are distinct. The weight of vertex v ∈ V ( G ) is defined to be the sum of the label of vertices in neighborhood of v and the label of all incident edges to v . The total distance vertex irregularity strength of G (denoted by t d i s ( G ) ) is the minimum of k for which G has a distance vertex irregular total k -labeling. In this paper, we present several results of the total distance vertex irregularity strength of some corona product graphs.