Incident Edges Research Articles

Dynamics of epidemic spreading on connected graphs.

We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connections between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a collection of numerical simulations for a selection of connected graphs.

Integer [formula omitted]-matchings of graphs: [formula omitted]-Berge–Tutte formula, [formula omitted]-factor-critical graphs and [formula omitted]-barriers

An integer k-matching of a graph G is a function h that assigns to each edge an integer in {0,…,k} such that ∑e∈Γ(v)h(e)≤k for each vertex v, where Γ(v) is the set of edges incident with v. The integer k-matching number of G is the maximum number of ∑e∈E(G)h(e) over all integer k-matchings h of G. When k is even, Yan Liu and Xiaohui Liu proved that the integer k-matching number of G equals k times its fractional matching number. In this paper, when k is odd, we prove the integer k-matching analogue of the Berge-Tutte Formula, define k-factor-critical graph, k-barrier and k-extreme, respectively, give two sufficient and necessary conditions of k-factor-critical graph which is similar to those of factor-critical graph, and obtain some properties about k-barrier and k-extreme, respectively.

(3,1) – total labeling of cycle, path graphs and some related graphs

The (3,1)- total labeling of a graph G is an assignment of non- negative integers to vertices and edges of G such that no two adjacent vertices or edges have the same label and the difference of labels between a vertex and its incident edges is at least 3. In this paper, we study the (3,1)- total labeling for some types of graphs and derive their (3,1)- total labeling number.

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

Partially normal 5-edge-colorings of cubic graphs

In a proper edge-coloring of a cubic graph, an edge e is normal if the set of colors used by the five edges incident with an end of e has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter μ3 is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph G has a proper 5-edge-coloring such that at least |E(G)|−μ3(G) (which is no less than 2735|E(G)|) edges are normal. This result improves on some earlier results of Bílková and Šámal.

Some Remarks on Odd Edge Colorings of Digraphs

The principal aim of this article is to initiate a study of the following coloring notion for digraphs. An odd k-edge coloring of a general digraph (directed pseudograph) D is a (not necessarily proper) coloring of its edges with at most k colors such that for every vertex v and color c holds: if c is used on the set ∂D(v) of edges incident with v, then c appears an odd number of times on each non-empty set from the pair ∂D+(v),∂D−(v) of, respectively, outgoing and incoming edges incident with v. We show that it can be decided in polynomial time whether D admits an odd 2-edge coloring. Throughout the paper, several conjectures, questions and open problems are posed. In particular, we conjecture that for each odd edge-colorable digraph four colors suffice.

Minimum Reload Cost Graph Factors

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to the problem of finding an r-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of r = 2, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of r and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT-algorithm.

Quasi-metrics on graphs

In this paper we consider some questions of the theory and practice of mean first passage time quasi-metric, a generalized metric structure closely related to ergodic homogeneous Markov chains. In particular, we consider the structure and properties of mean first passage time quasimetrics based on several graph models, including simple cycles, simple paths and their oriented analogues. The introduction contains the history of the problem and provides an overview of the main ideas and results presented in the article. The second section gives basic concepts of the theory of Markov chains. In fact, a Markov chain is a mathematical model of some random process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. This section collects basic definitions needed to consider the role of graph models in the presentation and study of ergodic homogeneous Markov chains. The Markov chain can be depicted as an oriented weighted graph of transitions whose vertices correspond to the states of the chain and the arcs correspond to the transitions between them. The process will be ergodic if this weighted oriented graph is weakly connected, and the largest common divisor of the lengths of all its cycles is equal to 1. On the other hand, any connected graph can be used as a basis for building a model of the simplest Markov chain: if a vertex 𝑖 has degree 𝑘, all incident edges are converted into arcs with the weights 1/𝑘 . Moreover, in the second section the definition of the mean first passage time for an ergodic homogeneous Markov chain is given. The algorithm of finding the mean first passage time is analyzed in detail by using converging trees of the oriented graph, related to the transition matrix of the ergodic homogeneous Markov chain. At last, a mean first passage time is analyzed as the quasi-metric 𝑚 of mean first passage time on the vertices 𝑉 = {1, 2, ..., 𝑛} of the oriented graph corresponding to the transition matrix of a given ergodic homogeneous Markov chain: 𝑚(𝑖, 𝑗) is the expected number of steps (arcs) for random wandering on the oriented graph Γ, starting at 𝑖, to reach 𝑗 for the first time. This quasi-metric has a number of important theoretical and applied properties. The third section deals with the construction and research of mean first passage time quasimetrics for the undirected cycles 𝐶𝑛, 𝑛 ≥ 3. Examples of constructions of mean first passage time quasi-metrics of undirected cycles for small values of 𝑛 are considered. Illustrations of the ”graphical“ procedure of building the matrix 𝑀 are given. Properties of the resulting generalized metric structures are analyzed. In the fourth section, similar considerations for mean first passage time quasi-metrics of the undirected paths 𝑃𝑛, 𝑛 ≥ 2, are represented. The fifth section deals with the construction and research of mean first passage time quasimetrics for the directed (oriented) cycles 𝐶𝑛, 𝑛 ≥ 3. Examples of constructions of mean first passage time quasi-metrics of undirected (oriented) cycles for small values of 𝑛 are considered. Illustrations of the ”graphical“ procedure of building the matrix 𝑀 are given. Properties of the resulting generalized metric structures are analyzed. In the sixth section, similar considerations for mean first passage time quasi-metrics of the directed (oriented) paths 𝑃𝑛, 𝑛 ≥ 2, are represented. In the conclusion, the results of the work are summed up and possible directions of further research are outlined.

Minimum Scan Cover with Angular Transition Costs

We provide a comprehensive study of a natural graph optimization problem that arises from transition costs between incident edges. In the problem Minimum Scan Cover with Angular Costs (MSC), we are...

Weakly reinforced Pólya urns on countable networks

We study the long-time asymptotics of a network of weakly reinforced Pólya urns. In this system, which extends the WARM introduced by R. van der Hofstad et. al. (2016) to countable networks, the nodes fire at times given by a Poisson point process. When a node fires, one of the incident edges is selected with a probability proportional to its weight raised to a power α<1, and then this weight is increased by 1. We show that for α<1∕2 on a network of bounded degrees, every edge is reinforced a positive proportion of time, and that the limiting proportion can be interpreted as an equilibrium in a countable network. Moreover, in the special case of regular graphs, this homogenization remains valid beyond the threshold α=1∕2.

Low dark current and high bandwidth evanescent wave coupled PIN photodetector array for 400 Gbit/s receiving system

Compared with surface and edge incident photodetectors, evanescent coupling photodetector (ECPD) has high bandwidth and high quantum efficiency, so it has a broad application prospect in the field of high-speed optical communication. The evanescent wave coupled photodetector is composed of a diluted waveguide, a single-mode ridge waveguide and a PIN photodiode. By directional evanescent wave coupling, the coupling efficiency of the incident light from the fiber to the absorption core of the photodetector is improved. In this paper, the structure design, experimental preparation and test results of an indium phosphorus based evanescent wave coupled photodetector array are introduced in detail. The test results show that the dark current of the evanescent wave coupled photodetector array is as low as 215 pA and 1.23 pA under –3 and 0 V bias, respectively. When the active area is 5 μm × 20 μm, the device still has a high responsivity of 0.5 A/W (without antireflection film). The high frequency performance of the detector is tested. The bandwidth of each detector is more than 25 GHz, and the total bandwidth is more than 400 GHz. Any optical device can be integrated. The detector array can be applied to the WDM receiving system of 400 Gbit/s and coherent receiving system of 200 Gbit/s.

A short proof of anti-Ramsey number for cycles

Ramsey's theorem states that there exists a least positive integer R(r, s) for which every blue-red edge colouring of the complete graph on R(r, s) vertices contains a blue clique on r vertices or a red clique on s vertices. This work contains a simplified proof of Anti-Ramsey theorem for cycles. If there is an edge e between H and H0, incident to, say, some v ∈ H of color from NEWc(v), then we can make H and H0 connected by adding the edge e and deleting some edge incident to v of the same color as e in H, so the resulting graph G˜ has a connected component of order ≥ 2( k+1/2 ), which contradicts that every connected component is of order ≤ k − 1. Since each component is Hamiltonian and of order ≥ k+1/ 2 , to avoid a rainbow Ck, by the same type of argument as in Claim 1, we must have that |c(H, H0 )| = 1.

EDGE ODD GRACEFUL LABELING OF SOME FLOWER PETAL GRAPHS

A labeling of a graph $G$ with $\alpha$ vertices and $\beta$ edges called an edge odd graceful labeling if there is an edge labeling with odd numbers to all edges such that each vertex is assigned a label which is the sum $\mod (2\gamma)$ of labels of edge incident on it, where $\gamma=max\{\alpha,\beta\}$ and the induced vertex labels are distinct. In this paper, we discussed about edge odd gracefulness of some special class of flower petal graphs.

Single-conflict colouring.

Given a multigraph, suppose that each vertex is given a local assignment of k colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least k for which this is always possible given any set of local assignments we call the single‐conflict chromatic number of the graph. This parameter is closely related to separation choosability and adaptable choosability. We show that single‐conflict chromatic number of simple graphs embeddable on a surface of Euler genus g is O(g1∕4log g) as g→∞. This is sharp up to the logarithmic factor.

Distributed Approximation Algorithms for Steiner Tree in the CONGESTED CLIQUE

The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGESTED CLIQUE model (CCM) [29] of distributed computing. For the Steiner tree problem in the CCM, we consider that each vertex of the input graph is uniquely mapped to a processor and edges are naturally mapped to the links between the corresponding processors. Regarding output, each processor should know whether the vertex assigned to it is in the solution or not and which of its incident edges are in the solution. We present two deterministic distributed approximation algorithms for the Steiner tree problem in the CCM. The first algorithm computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages for a given connected undirected weighted graph of [Formula: see text] nodes. Note here that [Formula: see text] notation hides polylogarithmic factors in [Formula: see text]. The second one computes a Steiner tree using [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] and [Formula: see text] are the shortest path diameter and number of edges respectively in the given input graph. Both the algorithms achieve an approximation ratio of [Formula: see text], where [Formula: see text] is the number of leaf nodes in the optimal Steiner tree. For graphs with [Formula: see text], the first algorithm exhibits better performance than the second one in terms of the round complexity. On the other hand, for graphs with [Formula: see text], the second algorithm outperforms the first one in terms of the round complexity. In fact when [Formula: see text] then the second algorithm achieves a round complexity of [Formula: see text] and message complexity of [Formula: see text]. To the best of our knowledge, this is the first work to study the Steiner tree problem in the CCM.

Very fast construction of bounded‐degree spanning graphs via the semi‐random graph process

In this paper, we study the following recently proposed semi‐random graph process: starting with an empty graph on n vertices, the process proceeds in rounds, where in each round we are given a uniformly random vertex v, and must immediately (in an online manner) add to our graph an edge incident with v. The end goal is to make the constructed graph satisfy some predetermined monotone graph property. Alon asked whether every given bounded‐degree spanning graph can be constructed with high probability in O(n) rounds. We answer this question positively in a strong sense, showing that any n‐vertex graph with maximum degree can be constructed with high probability in rounds. This is tight up to a multiplicative factor of . We also obtain tight bounds for the number of rounds necessary to embed bounded‐degree spanning trees, and consider a nonadaptive variant of this setting.

Double vertex-edge domination in graphs: complexity and algorithms

In a graph $$G=(V,E)$$ , a vertex $$v\in V$$ is said to ve-dominate the edges incident on v as well as the edges adjacent to these incident edges on v. A set $$D\subseteq V$$ is called a double vertex-edge dominating set if every edge of the graph is ve-dominated by at least two vertices of D. Given a graph G, the double vertex-edge dominating problem, namely Min-DVEDS is to find a minimum double vertex-edge dominating set of G. In this paper, we show that the decision version of Min-DVEDS is NP-complete for chordal graphs. We present a linear time algorithm to find a minimum double vertex-edge dominating set in proper interval graphs. We also show that for a graph having n vertices, Min-DVEDS cannot be approximated within $$(1 -\varepsilon ) \ln n$$ for any $$\varepsilon > 0$$ unless NP $$\subseteq $$ DTIME( $$n^{O(\log \log n)}$$ ). On positive side, we show that Min-DVEDS can be approximated by a factor of $$O(\ln \varDelta )$$ . Finally, we show that Min-DVEDS is APX-complete for graphs with maximum degree 5.

Vertices Belonging to all or to no Minimum Vertex-Edge Dominating Sets in Trees

Let G be a simple graph with vertex set V (G) and edge set E(G). A vertex v ∈ V (G) vertex-edge 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 vertex-edge dominated by a vertex of D. The vertex-edge dominating set of a minimum cardinality is called minimum vertex-edge dominating set. In this paper, we characterize the set of vertices that are in all or in no minimum vertex-edge dominating sets in trees.

Super Edge-Connected Linear Hypergraphs

A connected graph [Formula: see text] is super edge-connected, if every minimum edge-cut of [Formula: see text] is the set of edges incident with a vertex. In this paper, the concept of super edge-connectivity of graphs is generalized to hypergraphs and a necessary and sufficient condition for an [Formula: see text]-uniform and linear hypergraph with diameter at most 2 to be super edge-connected is given.

Note on diagonal construction of Z2nm-supermagic labeling of Cn□Cm

A Γ-supermagic labeling of a graph G(V, E) with is a bijection from E to an Abelian group Γ of order k such that the sum of labels of all incident edges of every vertex is equal to the same element An existence of a -supermagic labeling of Cartesian product of two cycles, for n odd was proved recently. This along with an earlier result by Ivančo proved the existence of a -supermagic labeling of for every In this paper we present a simple unified labeling method for all