Let G be a graph on n t> 3 vertices. Dirac's minimum degree condition is the condition that all vertices of G have degree at least in. This is a well-known sufficient condition for the existence of a Hamilton cycle in G. We give related sufficiency conditions for the existence of a Hamilton cycle or a perfect matching involving a restriction of Dirac's minimum degree condition to certain subset...

An Approximate Minimum Degree ordering algorithm (AMD) for preordering a symmetric sparse matrix prior to numerical factorization is presented. We use techniques based on the quotient graph for matrix factorization that allow us to obtain computationally cheap bounds for the minimum degree. We show that these bounds are often equal to the actual degree. The resulting algorithm is typically much...

Consider a directed graph G = (V,E) with n vertices and a root vertex r ∈ V . The DMDST problem for G is one of constructing a spanning tree rooted at r, whose maximal degree is the smallest among all such spanning trees. The problem is known to be NP-hard. A quasipolynomial time approximation algorithm for this problem is presented. The algorithm finds a spanning tree whose maximal degree is a...

— Dynamic virtual machine migration is a key technique technology in the cloud computing, an algorithm is proposed in this thesis, which is a dynamic resource scheduling based on minimum load imbalancing measurement. First of all, load balancer in scheduler judges the overload phenomenon; we need to migrate VM (Virtual Machine) on physical machine. Then, according to the records in load balanc...

For a fixed multigraph H with vertices w1, . . . , wm, a graph G is H-linked if for every choice of vertices v1, . . . , vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing wi (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs. Given a connected multigraph H with k edges and minimum degree at least two and n 7.5k, we...

We consider the MinGreedy strategy for Maximum Cardinality Matching. MinGreedy repeatedly selects an edge incident with a node of minimum degree. For graphs of degree at most ∆ we show that MinGreedy achieves approximation ratio at least ∆−1 2∆−3 in the worst case and that this performance is optimal among adaptive priority algorithms in the vertex model, which include many prominent greedy mat...

This paper describes an optimal triangulation algorithm for rectangles. We derive lower bounds on the maximum degree of triangulation, and show that our triangulation algorithm matches the lower bounds. Several important observations are also made, including a zig-zag condition that can verify whether a triangulation can minimizes the maximum degree to 4 or not. In addition, this paper identifi...

We write H → G if every 2-coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G-minimal if H → G, but for every proper subgraph H ′ of H, H ′ 6→ G. We define s(G) to be the minimum s such that there exists a G-minimal graph with a vertex of degree s. We prove that s(Kk) = (k − 1) and s(Ka,b) = 2 min(a, b) − 1. We also pose several related open problems. c © 2...

We answer a question of Zhao [SIAM J. Disc. Math. 23 vol.2, (2009), 888-900] that determines the minimum degree threshold for a bipartite graph G to contain an H-factor (a perfect tiling of G with H) for any bipartite graph H. We also show that this threshold is best possible up to a constant depending only on H. This result can be viewed as an analog to Kuhn and Osthus' result [Combinatorica 2...