Edge Uv Research Articles

Competitively orientable complete multipartite graphs

We say that a digraph D is competitive if any pair of vertices has a common out-neighbor in D and that a graph G is competitively orientable if there exists a competitive orientation of G. The competition graph of a digraph D is defined as the graph with the vertex set V(D) and an edge uv if and only if u and v compete in D. The notion of competitive digraph arose while studying digraph whose competition graphs are complete. We derive some useful properties of competitively orientable graphs and show that a complete graph of order n is competitively orientable if and only if n≥7. Then we completely characterize a competitively orientable complete multipartite graph in terms of the sizes of its partite sets. Moreover, we present a way to build a competitive multipartite tournament in each of competitively orientable cases.

Sum divisor cordial labeling in the context of graph operations on grötzsch

A Sum divisor cordial labeling of a graph G with vertex set V is a bijection r from V to {1,2,3,...,|V (G )|} such that an edge uv is assigned the label 1 if 2 divides r(u)+ r (v ) and 0 otherwise; and the number of edges labeled with 0 and the number of edges labeled with 1differ by at most 1 . A graph with a sum divisor cordial labeling is called sum divisor cordial graph. In this research paper, we investigate the sum divisor cordial labeling bahevior for Grötzsch graph, fusion of any two vertices in Grötzsch graph, duplication of an arbitrary vertex in Grötzsch graph, duplication of an arbitrary vertex by an edge in Grötzsch graph, switching of an arbitrary vertex of degree four in Grötzsch graph, switching of an arbitrary vertex of degree three in Grötzsch graph and path union of two copies of Grötzsch.

K-Product Cordial Behaviour of Union of Graphs

Let f be a map from V (G) to {0, 1, ..., k − 1} where k is an integer, 1 ≤ k ≤ |V (G)|. For each edge uv assign the label f(u)f(v)(mod k). f is called a k-product cordial labeling if |vf (i) − vf (j)| ≤ 1, and |ef (i) − ef (j)| ≤ 1, i, j ∈ {0, 1, ..., k − 1}, where vf (x) and ef (x) denote the number of vertices and edges respectively labeled with x (x = 0, 1, ..., k − 1). In this paper, we investigate the k-product cordial behaviour of union of graphs

Star colouring of bounded degree graphs and regular graphs

A k-star colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that f(u)≠f(v) for every edge uv of G, and every bicoloured connected subgraph of G is a star. The star chromatic number of G, χs(G), is the least integer k such that G is k-star colourable. We prove that χs(G)≥⌈(d+4)/2⌉ for every d-regular graph G with d≥3. We reveal the structure and properties of even-degree regular graphs G that attain this lower bound. The structure of such graphs G is linked with a certain type of Eulerian orientations of G. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for p≥2, a 2p-regular graph G is (p+2)-star colourable only if n:=|V(G)| is divisible by (p+1)(p+2). For each p≥2 and n divisible by (p+1)(p+2), we construct a 2p-regular Hamiltonian graph on n vertices which is (p+2)-star colourable.The problem k-Star Colourability takes a graph G as input and asks whether G is k-star colourable. We prove that 3-Star Colourability is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For k≥3, k-Star Colourability of bipartite graphs of maximum degree k is NP-complete, and does not even admit a 2o(n)-time algorithm unless ETH fails.

Strong edge coloring of circle graphs

A strong clique in a graph G is a set S of edges of G such that every two edges in S are either adjacent or joined by an edge of G. We say that G is a circle graph if vertices of G correspond to chords of a circle and G contains an edge uv if and only if chords corresponding to u and v intersect.We show that the maximum size of a strong clique in a circle graph with maximum degree Δ is at most 54Δ2. This bound is tight for even Δ and correct up to a smaller order error term for odd Δ.This result supports a conjecture of Erdős and Nešetřil from 1985, which states that the strong chromatic index of a graph with maximum degree Δ is at most 54Δ2. It also confirms a conjecture of Faudree, Gyárfás, Schelp and Tuza from 1990 for circle graphs.

Module Basis for Generalized Spline Modules

Let G = (V,E) be a graph of order n. Let R be a commutative ring and I denote the set of all ideals of R. Let ? : E ? I be an edge labeling. A generalized spline of (G, ?) is a vertex labeling F : V ? R such that for each edge uv, F(u) ? F(v) ? ?(uv). The set R(G,) of all generalized splines of (G, ?) is an R-module. In this paper we determine conditions for a subset of R(G,?) to form a basis of R(G,?) for some classes of graphs.

Proper distinguishing arc-colourings of symmetric digraphs

A symmetric digraph G↔ arises from a simple graph G by substituting each edge uv by a pair of opposite arcs uv→,vu→. An arc-colouring c of G↔ is distinguishing if the only automorphism of G↔ preserving c is the identity. We study four types of proper arc-colourings of G↔ corresponding to four definitions of adjacency of arcs. For each type, we investigate the distinguishing chromatic index of G↔, i.e. the least number of colours in a distinguishing proper colouring of G↔. We also determine tight bounds for chromatic indices of G↔, i.e. for the least numbers of colours in each type of proper colourings. Colourings of arcs of a symmetric digraph G↔ are equivalent to colourings of halfedges of the graph G, which have applications in computer science.

On some graph-cordial Abelian groups

Hovey introduced A-cordial labelings as a generalization of cordial and harmonious labelings [11]. If A is an Abelian group, then a labeling f:V(G)→A of the vertices of some graph G induces an edge labeling on G; the edge uv receives the label f(u)+f(v). A graph G is A-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one.In the literature mostly a cordial labeling in cyclic groups is studied. Patrias and Pechenik studied the larger class of finite Abelian groups A. They posed a conjecture that for every group A there is an A-cordial labeling for almost every path. In this paper we solve this conjecture. Moreover we show that all cycle graphs are A-cordial for any Abelian group A of odd order.

Extremal augmented Zagreb index of trees with given numbers of vertices and leaves

The augmented Zagreb index AZI(G) of a connected graph G is defined asAZI(G)=∑uv∈E(G)(d(u)d(v)d(u)+d(v)−2)3, where d(u) and d(v) are the degrees of the end-vertices of an edge uv, respectively. We determine the unique tree with given numbers of vertices and leaves that minimizes augmented Zagreb index, and characterize the unique graph with minimum augmented Zagreb index in the class of connected graphs with given numbers of vertices and pendent vertices. Furthermore, we also determine the maximum augmented Zagreb index with the extremal trees characterized in the class of all trees with given numbers of vertices and leaves.

Vertex cover and Edge vertex domination in trees

Let G = (V,E) be a simple graph. An edge e ∈ E(G) edge-vertex dominates a vertex v ∈ V (G) if e is incident with v or e is incident with a vertex adjacent to v. A subset D ⊆ E(G) is an edge-vertex dominating set of a graph G if every vertex of G is edge-vertex dominated by an edge of D. A vertex cover of G is a set C ⊆ V such that for each edge uv ∈ E at least one of u and v is in C. We characterize trees with edge-vertex domination number equals vertex covering number.

Hamilton cycles in the semi-random graph process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on n vertices. In each round, a vertex u is presented to the player independently and uniformly at random. The player then adaptively selects a vertex v, and adds the edge uv to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible.We focus on the problem of constructing a Hamilton cycle in as few rounds as possible. In particular, we present a novel strategy for the player which achieves a Hamiltonian cycle in c∗n rounds, where the value of c∗ is the result of a high dimensional optimization problem. Numerical computations indicate that c∗<2.61135. This improves upon the previously best known upper bound of 3n rounds. We also show that the previously best lower bound of (ln2+ln(1+ln2)+o(1))n is not tight.

On the cordial labeling of certain trigraphs

Let G be a graph that has n vertices and m edges. Let f: V(G) → {1,2,..., k} be a function that assigns to each vertex v ∈ G a positive integer f(v) ∈ {1,2,..., k}. We assign to each edge uv ∈ E(G) a label which is the gcd(f(u), f(v)). The function f is called k-prime cordial labeling of G if |v f (i) − v f (j)| ≤ 1 for all i, j ∈ {1,2,..., k} and |e f (0) − e f (1)| ≤ 1, where v f (i) denotes the number of vertices labeled with i, e f (1) and e f (0) denote the number of edges labeled with 1 and not labeled with 1, respectively. In this paper, we introduce the concept of trigraph of a graph G, T3(G), and we show that the trigraph of a path P n , T 3 (P n ), and the trigraph of a cycle C n , T 3 (C n ) are 4-prime cordial graphs.

Linear-size planar Manhattan network for convex point sets

Let G=(V,E) be an edge weighted geometric graph (not necessarily planar) such that every edge is horizontal or vertical. The weight of an edge uv∈E is the L1-distance between its endpoints. Let WG(u,v) denotes the length of a shortest path between a pair of vertices u and v in G. The graph G is said to be a Manhattan network for a given point set P in the plane if P⊆V and ∀p,q∈P, WG(p,q)=‖pq‖1. In addition to P, the graph G may also include a set T of Steiner points in its vertex set V. In the Manhattan network problem, the objective is to construct a Manhattan network of small size (the number of Steiner points) for a set of n points. This problem was first considered by Gudmundsson et al. [EuroCG 2007]. They give a construction of a Manhattan network of size Θ(nlog⁡n) for general point sets in the plane. We say a Manhattan network is planar if it has a planar embedding. In this paper, we construct a planar Manhattan network for convex point sets in linear time using O(n) Steiner points. We also show that, even for convex point sets, the construction in Gudmundsson et al. [EuroCG 2007] needs Ω(nlog⁡n) Steiner points, and the network may not be planar.

Prime Cordial Labeling of Some Graphs

A bijectionfrom the vertex set V of a graph G to {1,2,…|V|} is called a prime cordial labeling of G if each edge uv is assigned the label 1 if gcd (f (u), f (v)) = 1 and 0 if gcd (f (u), f (v)) > 1, where the number of edges labeled with 0 and the number of edges labeled with 1 differ at most by one. In this paper we prove that 8-polygonal snake containing n number of 8-polygon, Splitting graph of Cn for n ≥ 5 and Armed Crown C2k Pm for all k ≥ 3 and m ≥ 2 admit prime cordial labeling.

A case study of an edge-magic total labeling of (a,b)-cycle books

Suppose α and β be the order and size of a graph G respectively. A one-one function h which maps the set of vertices and edges of a graph G onto the integers 1, 2, 3, …, α + β such that h(u) + h(uv) + h(v) = c for any edge (uv) ε E(G) is called an edge-magic total labeling of If h(u) ε {1,2, …, α} for any uεV(G) then h is a super edge-magic total labeling of G. One of interesting research topic is super edge-magic total labeling of cycle book. A cycle book B(a, m, b, n, t) is made up from m copies of cycle Ca and n copies cycle Cb with a commont path Pt . Super edge-magic total labeling of a cycle book B(a, m, b, n, 2) is still under investigation even for the case a = b This paper talk about a partial solution to this problem. We prove that a cycle book B(5, 2, 3, n, 2) has a super edge-magic total labeling for all positive integr n. In addition, we show that a cycle book B(5, 2, 3, n, 2) have an edge-magic total labeling for an integer n, 1 ≥ n ≥ 6.

SD-Prime cordial labeling of alternate k-polygonal snake of various types

Let f : V (G) → {1, 2,..., |V (G)|} be a bijection, and let us denote S = f(u) + f(v) and D = |f(u) − f(v)| for every edge uv in E(G). Let f' be the induced edge labeling, induced by the vertex labeling f, defined as f' : E(G) → {0, 1} such that for any edge uv in E(G), f' (uv)=1 if gcd(S, D)=1, and f' (uv)=0 otherwise. Let ef' (0) and ef' (1) be the number of edges labeled with 0 and 1 respectively. f is SD-prime cordial labeling if |ef' (0) − ef' (1)| ≤ 1 and G is SD-prime cordial graph if it admits SD-prime cordial labeling. In this paper, we have discussed the SD-prime cordial labeling of alternate k-polygonal snake graphs of type-1, type-2 and type-3.

Graceful Labeling and Skolem Graceful Labeling on the U-star Graph and It’s Application in Cryptography

Graceful Labeling on graph G=(V, E) is an injective function f from the set of the vertex V(G) to the set of numbers {0,1,2,...,|E(G)|} which induces bijective function f from the set of edges E(G) to the set of numbers {1,2,...,|E(G)|} such that for each edge uv e E(G) with u,v e V(G) in effect f(uv)=|f(u)-f(v)|. Meanwhile, the Skolem graceful labeling is a modification of the Graceful labeling. The graph has graceful labeling or Skolem graceful labeling is called graceful graph or Skolem graceful labeling graph. The graph used in this study is the U-star graph, which is denoted by U(Sn). The purpose of this research is to determine the pattern of the graceful labeling and Skolem graceful labeling on graph U(Sn) apply it to cryptography polyalphabetic cipher. The research begins by forming a graph U(Sn) and they are labeling it with graceful labeling and Skolem graceful labeling. Then, the labeling results are applied to the cryptographic polyalphabetic cipher. In this study, it is found that the U(Sn) graph is a graceful graph and a Skolem graceful graph, and the labeling pattern is obtained. Besides, the labeling results on a graph it U(Sn) can be used to form a table U(Sn) polyalphabetic cipher. The table is used as a key to encrypt messages.

Odd harmonious labeling of Sn (m, r) graph

A graph labeling is an assignment of integers to vertices or edges of a graph subject to certain conditions. There are various kinds of graph labeling, one of them is an odd harmonious labeling. An odd harmonious labeling f of a graph G on q edges is an injective function f from the set of vertices of G to the set {0,1,2,…,2q - 1} such that the induced function f*, where f* (uv) = f(u) + f(v) for every edge uv of G, is a bijection from the set of edges of G to {1,3,5,…,2q - 1}. A graph is said to be odd harmonious if it admits an odd harmonious labeling. A graph Sn(m, r) is a graph formed from r stars, each of which has n + 1 vertices, and every center of the star is joined to one new vertex v 0 by a path of length m. In this paper we show that the graph Sn (m, r), m ≥ 2, 1≤ r≤3, is odd harmonious.

Observation of monochromatic and coherent luminescence from nanocavities of GaN nanowall network

Scaling-down the size of semiconductor cavity lasers and engineering their electromagnetic environment in the Purcell regime can bring about spectacular advance in nanodevices fabrication. We report here an unprecedented observation of a coherent Cathodoluminescence from GaN nanocavities (20–100 nm). Incident lower energy (< 15 kV) electron beams excite the band edge UV emission from the walls of the network whereas for higher energies, the emitted photons are spontaneously down converted into NIR and preferentially emerge from the nanocavities. Non-centrosymmetric structure of GaN and its nanowall geometry together facilitate this unique observation which is substantiated by our numerical results. At cryogenic temperatures, an intense and narrow laser-like NIR beam emanates out of the nanocavities. The work promises the possibility of fabrication of very high density (over 108/cm2) cavity lasers that are addressable by simple deflection and tuning of incident electron beams.

Total weight choosability for Halin graphs

A proper total weighting of a graph G is a mapping φ which assigns to each vertex and each edge of G a real number as its weight so that for any edge uv of G , Σ e ∈ E ( v ) φ( e )+φ( v ) ≠ Σ e ∈ E ( u ) φ( e )+φ( u ). A ( k,k ')-list assignment of G is a mapping L which assigns to each vertex v a set L ( v ) of k permissible weights and to each edge e a set L ( e ) of k ' permissible weights. An L -total weighting is a total weighting φ with φ( z ) ∈ L ( z ) for each z ∈ V ( G ) ∪ E ( G ). A graph G is called ( k,k ')-choosable if for every ( k,k ')-list assignment L of G , there exists a proper L -total weighting. As a strenghtening of the well-known 1-2-3 conjecture, it was conjectured in [Wong and Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph without isolated edge is (1,3)-choosable. It is easy to verified this conjecture for trees, however, to prove it for wheels seemed to be quite non-trivial. In this paper, we develop some tools and techniques which enable us to prove this conjecture for generalized Halin graphs.