The formula for Turán number of spanning linear forests

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.

Covering Non-uniform Hypergraphs

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.

Extremal graphs for blow-ups of stars and paths

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.

Constructing extremal triangle-free graphs using integer programming

In 1959 Erd\H os and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. We investigate a rainbow version of the theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$.

Minimum number of edges in graphs that are both P2- and Pi-connected

The Hierarchical Chinese postman problem is a special type of Chinese postman problem. The aim is to find a shortest tour that traverses each edge of a given graph at least once. The constraint is that the arcs are partitioned into classes and a precedence relation orders the classes according to priority. Different forms of the HCPP are applied in real life applications such as snow plowing, winter gritting and street sweeping. The problem is solvable in polynomial time if the ordering relation is linear and each class is connected. Dror et al. (1987) presented an algorithm which provides time complexity of O (kn <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> ). CPP which is lower bound for HCPP. We give alternate approach by using Kruskal's method to reduce number of edges in graph which is having time complexity of O (k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), where k is number of layers in graph and n is number of nodes in graph. It is found that the suggested kruskal-based HCPP-solution gives average 21.64% improvement compare to simple HCPP and we get average 13.35% improvement over CPP when number of hierarchy is less than 3 and numbers of edges in graph are less than 10.

Let k ≥ 1 k \geq 1 be an odd integer, t = ⌊ k + 2 4 ⌋ {t = \left \lfloor {\tfrac {{k + 2}}{4}} \right \rfloor } , and q be a prime power. We construct a bipartite, q-regular, edge-transitive graph C D ( k , q ) CD(k,q) of order υ ≤ 2 q k − t + 1 \upsilon \leq 2{q^{k - t + 1}} and girth g ≥ k + 5 g \geq k + 5 . If e is the the number of edges of C D ( k , q ) CD(k,q) , then e = Ω ( υ 1 + 1 k − t + 1 ) e = \Omega ({{\upsilon ^{1 + \frac {1}{{k - t + 1}}}}}) . These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order υ \upsilon and girth at least g, g ≥ 5 g \geq 5 , g ≠ 11 g \ne 11 , 12. For g ≥ 24 g \geq 24 , this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for 5 ≤ g ≤ 23 5 \leq g \leq 23 , g ≠ 11 g \ne 11 , 12, it improves on or ties existing bounds.

Bounds on the size of graphs of given order and l-connectivity

On the number of edges in graphs with a given weakly connected domination number

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

On unique independent sets in graphs

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.