The number of edges in graphs with bounded clique number and circumference

The study of cyclic graphs has always been a hot topic in the field of graph theory and has received widespread attention from graph theory practitioners. If an n-order graph G exactly contains cycles of all lengths from 3 to n, then the graph is called a pan cycle graph. This article proves that, after excluding some special cases, when the number of edges in graph G is greater than or equal to C2n-3+12, graph G must be a pan cyclic graph.

Constructing extremal triangle-free graphs using integer programming

Almost optimal query algorithm for hitting set using a subset query

The image segmentation of histopathological tissue images has always been a challenge due to the overlapping of tissue color distributions, the complexity of extracellular texture and the large image size. In this paper, we introduce a new region-merging algorithm, namely, the Regional Pattern Merging (RPM) for interactive color image segmentation and annotation, by efficiently retrieving and applying the user’s prior knowledge of stroke-based interaction. Low-level color/texture features of each region are used to compose a regional pattern adapted to differentiating a foreground object from the background scene. This iterative region-merging is based on a modified Region Adjacency Graph (RAG) model built from initial segmented results of the mean shift to speed up the merging process. The foreground region of interest (ROI) is segmented by the reduction of the background region and discrimination of uncertain regions. We then compare our method against state-of-the-art interactive image segmentation algorithms in both natural images and histological images. Taking into account the homogeneity of both color and texture, the resulting semi-supervised classification and interactive segmentation capture histological structures more completely than other intensity or color-based methods. Experimental results show that the merging of the RAG model runs in a linear time according to the number of graph edges, which is essentially faster than both traditional graph-based and region-based methods.

Extremal graphs for blow-ups of stars and paths

We study the task of estimating the number of edges in a graph, where the access to the graph is provided via an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n -vertex graph, using (i) polylog( n ) bipartite independent set queries or (ii) n 2/3 polylog( n ) independent set queries.

The formula for Turán number of spanning linear forests

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known practical algorithms for computing the hyperbolicity number of a n-vertex graph have running time \(O(n^4)\). Exploiting the framework of parameterized complexity analysis, we explore possibilities for “linear-time FPT” algorithms to compute hyperbolicity. For instance, we show that hyperbolicity can be computed in time \(2^{O(k)} + O(n +m)\) (m being the number of graph edges, k being the size of a vertex cover) while at the same time, unless the SETH fails, there is no \(2^{o(k)}n^2\)-time algorithm.

A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)|=k for all v, and a graph G is k-list-critical if G is not (k−1)-choosable, but every proper subgraph of G is (k−1)-choosable. The problem of determining the minimum number of edges in a k-critical graph with n vertices has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best known lower bound on the number of edges in a k-list-critical graph. In fact, our result on k-list-critical graphs is derived from a lower bound on the number of edges in a graph with Alon–Tarsi number at least k. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.

Due to a calculation error, the constant in the main theorem is not $80 \sqrt{k\log k}$ but $80\sqrt{k} \log k$. The error was discovered by Xizhi Liu.For a new discussion on the limit of the method used in this paper see the revised arXiv version of the paper.

The bipartite graph is a ubiquitous data structure that can model the relationship between two entity types: for instance, users and items, queries and webpages. In this paper, we study the problem of ranking vertices of a bipartite graph, based on the graph's link structure as well as prior information about vertices (which we term a query vector). We present a new solution, BiRank, which iteratively assigns scores to vertices and finally converges to a unique stationary ranking. In contrast to the traditional random walk-based methods, BiRank iterates towards optimizing a regularization function, which smooths the graph under the guidance of the query vector. Importantly, we establish how BiRank relates to the Bayesian methodology, enabling the future extension in a probabilistic way. To show the rationale and extendability of the ranking methodology, we further extend it to rank for the more generic n-partite graphs. BiRank's generic modeling of both the graph structure and vertex features enables it to model various ranking hypotheses flexibly. To illustrate its functionality, we apply the BiRank and TriRank (ranking for tripartite graphs) algorithms to two real-world applications: a general ranking scenario that predicts the future popularity of items, and a personalized ranking scenario that recommends items of interest to users. Extensive experiments on both synthetic and real-world datasets demonstrate BiRank's soundness (fast convergence), efficiency (linear in the number of graph edges) and effectiveness (achieving state-of-the-art in the two real-world tasks).

We show that, for each fixed k, an n-vertex graph not containing a cycle of length 2k has at most $80\sqrt{k\log k}\cdot n^{1+1/k}+O(n)$ edges.

On minimum average stretch spanning trees in polygonal 2-trees

Known asymptotic formulas are used to analyze the connectivity probability of graphs with highly reliable and poorly reliable edges. The conventional methods used to calculate the coefficients in these formulas require a number of arithmetic operations that grows geometrically with the growing number of graph edges. The adjacency matrix of the dual graph for highly reliable edges and the Kirchhoff matrix for poorly reliable edges result in algorithms of cubic complexity.

Given a hypergraph, a partition of its vertex set, and a nonnegative integer k, find a minimum number of graph edges to be added between different members of the partition in order to make the hypergraph k-edge-connected. This problem is a common generalization of the following two problems: edge-connectivity augmentation of graphs with partition constraints (J. Bang-Jensen, H. Gabow, T. Jordan, Z. Szigeti, SIAM J Discrete Math 12(2) (1999), 160–207) and edge-connectivity augmentation of hypergraphs by adding graph edges (J. Bang-Jensen, B. Jackson, Math Program 84(3) (1999), 467–481). We give a min–max theorem for this problem, which implies the corresponding results on the above-mentioned problems, and our proof yields a polynomial algorithm to find the desired set of edges.

In this article, we consider 'N'spherical caps of area 4πp were uniformly distributed over the surface of a unit sphere. We study the random intersection graph G N constructed by these caps. We prove that for p = c N α ,c >0 and α> 2, the number of edges in graph G N follow the Poisson distribution. I. I NTRODUCTION . Sensor network pose a number of challenging problems such as coverage, connectivity and tracking. We are considering a typical network consisting two major physical components. First, we have station. A station is an endpoint of connection with a wireless interface used to the network. Typical examples of stations are laptop, mobile etc. Second the access An point has one wireless interface and one wired interface. The wired interface is effective between different points and the wireless interface is effective between the stations and the point. It is therefore the point that connects the wireless LAN to the wired LAN. Also the radio propagation effects limit the range of wireless transmission. This range can be increased by increasing the transmission power.

A number of modeling and simulation algorithms using internal coordinates rely on hierarchical representations of molecular systems. Given the potentially complex topologies of molecular systems, though, automatically generating such hierarchical decompositions may be difficult. In this article, we present a fast general algorithm for the complete construction of a hierarchical representation of a molecular system. This two-step algorithm treats the input molecular system as a graph in which vertices represent atoms or pseudo-atoms, and edges represent covalent bonds. The first step contracts all cycles in the input graph. The second step builds an assembly tree from the reduced graph. We analyze the complexity of this algorithm and show that the first step is linear in the number of edges in the input graph, whereas the second one is linear in the number of edges in the graph without cycles, but dependent on the branching factor of the molecular graph. We demonstrate the performance of our algorithm on a set of specifically tailored difficult cases as well as on a large subset of molecular graphs extracted from the protein data bank. In particular, we experimentally show that both steps behave linearly in the number of edges in the input graph (the branching factor is fixed for the second step). Finally, we demonstrate an application of our hierarchy construction algorithm to adaptive torsion-angle molecular mechanics.

Augmenting the edge-connectivity of a hypergraph by adding a multipartite graph

On mutually independent hamiltonian paths