On mutually independent hamiltonian paths

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P 1=〈u 1,u 2,…,u n 〉 and P 2=〈v 1,v 2,…,v n 〉 of G are said to be independent if u 1=v 1, u n =v n , and u i ≠v i for all 1<i<n; and both are full-independent if u i ≠v i for all 1≤i≤n. Moreover, P 1 and P 2 are independent starting at u 1, if u 1=v 1 and u i ≠v i for all 1<i≤n. A set of Hamiltonian paths {P 1,P 2,…,P k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u 1) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u 1). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q n . In this paper, we show the following results: When |F|≤n−4, Q n −F−{x,y} remains Hamiltonian-laceable, where x and y are any two vertices from different partite sets and n≥4. When |F|≤n−2, Q n −F contains (n−|F|−1)-pairwise full-independent Hamiltonian paths between n−|F|−1 pairs of adjacent vertices, where n≥2. When |F|≤n−2, Q n −F contains (n−|F|−1)-pairwise independent Hamiltonian paths starting at any vertex v in a partite set to n−|F|−1 distinct vertices in the other partite set, where n≥2. When 1≤|F|≤n−2, Q n −F contains (n−|F|−1)-pairwise independent Hamiltonian paths between any two vertices from different partite sets, where n≥3.

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.

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

The formula for Turán number of spanning linear forests

Covering Non-uniform Hypergraphs

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

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

In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C(u,v) be a condition on two nonadjacent vertices u and v of a graph G. Then G+uv has property P if and only if G has property P. The first and now well-known result of this type was established by Bondy and Chvatal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G+uv is hamiltonian if and only if G is hamiltonian. Based on this result, they defined the n-closure cln (G) of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cln(G) is well-defined, and that G is hamiltonian if and only if cln(G) is hamiltonian. Moreover, they showed that cln(G) can be obtained by a polynomial algorithm, and that a Hamilton cycle in cln(G) can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cln(G)=Kn (and n≥3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic sufficient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their first paper on closures, Bondy and Chvatal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their first results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years.

On the mutually independent Hamiltonian cycles in faulty hypercubes

We study graph theory. The graphs we consider in the thesis consist of a finite vertex set and a finite edge set, and each edge joins an unordered pair of (not necessarily distinct) vertices. In a graph, a round trip which visits a subset of the vertices once, is called a cycle. If such a round trip passes through every vertex of the graph precisely once, we call it a Hamilton cycle. The Hamilton problem, which is the problem of determining whether a Hamilton cycle exists in a given graph has been proved to be NP-complete. So most researchers are focussing on establishing sufficient conditions, i.e., conditions on the graph that guarantee the existence of a Hamilton cycle. In this thesis, we focus on giving sufficient conditions for hamiltonian properties of a graph in terms of structural or algebraic parameters. Besides the Hamilton cycle, we also consider traceability. It requires that there exists a Hamilton path in the graph, i.e., a path through the vertices and edges of the graph that visits all the vertices exactly once (and does not return to the starting vertex). We say a graph is traceable if it contains a Hamilton path, and traceable from a vertex x if it contains a Hamilton x-path, i.e., a Hamilton path starting at vertex x. So, the next hamiltonian property we deal with is that the graph is traceable from every arbitrary vertex. The last hamiltonian property we take into account is Hamilton-connectivity, which requires that any two distinct vertices can be connected by a Hamilton path. The results in this thesis mainly involve degree conditions, spectral conditions, and Wiener index and Harary index conditions for traceability, hamiltonicity or Hamilton-connectivity of graphs. All of our results involve the characterization of the exceptional graphs, i.e., all the nonhamiltonian graphs that satisfy the condition. Our contributions extend existing results.

In 2005, Rahman and Kaykobad proved that if G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n+1 for each pair of distinct non-adjacent vertices u,v in G, then G is traceable [ Information Processing Letters, 94(2005), 1, 37-41]. In 2006, Li proved thatif G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n+3 for each pair of distinct non-adjacent vertices u,v in G, then G is Hamiltonian-connected [Information Processing Letters, 98(2006), 4, 159-161]. In this present paper, we prove that if G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n for each pair of distinct non-adjacent vertices u,v in G, then G has a Hamiltonian path or G belongs to a class of exceptional graphs. We also prove that if G is a 2-connected graph with n vertices and d(u)+d(v)+((u,v)≥n+2 for each pair of distinct non-adjacent vertices u,v in G, then G is Hamiltonian-connected or G belongs to a classes of exceptional graphs. Thus, our the two results generalize the above two results by Rahman et al. and Li, respectively.

LetM be anm-by-n matrix with entries in {0,1,⋯,K}. LetC(M) denote the minimum possible number of edges in a directed graph in which (1) there arem distinguished vertices calledinputs, andn other distinguished vertices calledoutputs; (2) there is no directed path from an input to another input, from an output to another output, or from an output to an input; and (3) for all 1 ≤i ≤m and 1 ≤j ≤n, the number of directed paths from thei-th input to thej-th output is equal to the (i,j)-th entry ofM. LetC(m,n,K) denote the maximum ofC(M) over allm-by-n matricesM with entries in {0,1,⋯,K}. We assume (without loss of generality) thatm ≥n, and show that ifm=(K+1) 0(n) andK=22 0(m) , thenC(m,n,K)= H/logH + 0(H/logH), whereH=mnlog(K + 1) and all logarithms have base 2. The proof involves an interesting problem of Diophantine approximation, which is solved by means of an unusual continued fraction expansion.

Spectra of quasi-strongly regular graphs