Xingxing Yu

4-Connected Projective-Planar Graphs Are Hamiltonian

We prove the result stated in the title (conjectured by Grunbaum) and a conjecture of Plummer that every graph which can be obtained from a 4-connected planar graph by deleting two vertices is Hamiltonian. The proofs are constructive and give rise to polynomial-time algorithms.

Ulysses: a robust, low-diameter, low-latency peer-to-peer network†

A number of distributed hash table (DHT)-based protocols have been proposed to address the issue of scalability in peer-to-peer networks. In this paper, we present Ulysses, a peer-to-peer network based on the butterfly topology that achieves the theoretical lower bound of log n/ log log n on network diameter when the average routing table size at nodes is no more than log n. Compared to existing DHT-based schemes with similar routing table size, Ulysses reduces the network diameter by a factor of log log n, which is 2–4 for typical configurations. This translates into the same amount of reduction on query latency and average traffic per link/node. In addition, Ulysses maintains the same level of robustness in terms of routing in the face of faults and recovering from graceful/ungraceful joins and departures, as provided by existing DHT-based schemes. The performance of the protocol has been evaluated using both analysis and simulation. Copyright

Finding Four Independent Trees

Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2-connected graph. Cheriyan and Maheshwari gave an

Disjoint paths, planarizing cycles, and spanning walks

O(|V|^2)

Covering planar graphs with forests

algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an

Ulysses: a robust, low-diameter, low-latency peer-to-peer network

O(|V|^3)

Long cycles in 3-connected graphs

algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs.

Five-Connected Toroidal Graphs Are Hamiltonian

We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected triangulation of a surface with sufficiently large representativity is hamiltonian, thus verifying a conjecture of Thomassen. We also obtain results about spanning walks in graphs embedded in a surface with large representativity.

Graph Connectivity After Path Removal

We study the problem of covering graphs with trees and a graph of bounded maximum degree. By a classical theorem of Nash-Williams, every planar graph can be covered by three trees. We show that every planar graph can be covered by two trees and a forest, and the maximum degree of the forest is at most 8. Stronger results are obtained for some special classes of planar graphs.

Independence ratios of graph powers

A number of distributed hash table (DHT)-based protocols have been proposed to address the issue of scalability in peer-to-peer networks. In this paper, we present Ulysses, a peer-to-peer network based on the butterfly topology that achieves the theoretical lower bound of (log n)/(log log n)on network diameter when the average routing table size at nodes is no more than log n. Compared to existing DHT-based schemes with similar routing table size, Ulysses reduces the network diameter by a factor of log log n. which is 2-4 for typical configurations. This translates into the same amount of reduction on query latency and average traffic per link/node. In addition, Ulysses maintains the same level of robustness in terms of routing in the face of faults and recovering from graceful/ungraceful joins and departures, as provided by existing DHT-based schemes. The performance of the protocol has been evaluated using both analysis and simulation.