Decomposing dense bipartite graphs into 4-cycles

Let G be an even bipartite graph with partite sets X and Y such that |Y | is even and the minimum degree of a vertex in Y is at least 95|X|/96. Suppose furthermore that the number of edges in G is divisible by 4. Then G decomposes into 4-cycles

Cycles and Bases of Graphs and Matroids

The objective of this dissertation is to investigate the properties of cycles and bases in matroids and in graphs. In [62], Tutte defined the circuit graph of a matroid and proved that a matroid is connected if and only if its circuit graph is connected. Motivated by Tutte\u27s result, we introduce the 2nd order circuit graph of a matroid, and prove that for any connected matroid M other than U1,1, the second order circuit graph of M has diameter at most 2 if and only if M does not have a restricted minor isomorphic to U2,6.;Another research conducted in this dissertation is related to the eulerian subgraph problem in graph theory. A graph G is eulerian if G is connected without vertices of odd degrees, and G is supereulerian if G has a spanning eulerian subgraph. In [3], Boesch, Suffey and Tindel raised a problem to determine when a graph is supereulerian, and they remarked that such a problem would be a difficult one. In [55], Pulleyblank confirmed the remark by showing that the problem to determine if a graph is supereulerian, even within planar graphs, is NP-complete. Catlin in [8] introduced a reduction method based on the theory of collapsible graphs to search for spanning eulerian subgraphs in a given graph G. In this dissertation, we introduce the supereulerian width of a graph G, which generalizes the concept of supereulerian graphs, and extends the supereulerian problem to the supereulerian width problem in graphs. Further, we also generalize the concept of collapsible graphs to s-collapsible graphs and develop the reduction method based on the theory of s-collapsible graphs. Our studies extend the collapsible graph theory of Catlin. These are applied to show for any integer n \u3e 2, the complete graph Kn is (n - 3)- collapsible, and so the supereulerian width of Kn is n - 2. We also prove a best possible degree condition for a simple graph to have supereulerian width at least 3.;The number of edge-disjoint spanning trees plays an important role in the design of networks, as it is considered as a measure of the strength of the network. As disjoint spanning trees are disjoint bases in graphic matroids, it is important to study the properties related to the number of disjoint bases in matroids. In this dissertation, we develop a decomposition theory based on the density function of a matroid, and prove a decomposition theorem that partitions the ground set of a matroid M into subsets based on their densities. As applications of the decomposition theorem, we investigate problems related to the properties of disjoint bases in a matroid. We showed that for a given integer k \u3e 0, any matroid M can be embedded into a matroid M\u27 with the same rank (that is, r(M) = r( M\u27)) such that M\u27 has k disjoint bases. Further we determine the minimum value of |E( M\u27)| -- |E(M)| in terms of invariants of M. For a matroid M with at least k disjoint bases, we characterize the set of elements in M such that removing any one of them would still result in a matroid with at least k disjoint bases

Spanning Trails and Spanning Trees

There are two major parts in my dissertation. One is based on spanning trail, the other one is comparing spanning tree packing and covering.;The results of the spanning trail in my dissertation are motivated by Thomassen\u27s Conjecture that every 4-connected line graph is hamiltonian. Harary and Nash-Williams showed that the line graph L( G) is hamiltonian if and only if the graph G has a dominating eulerian subgraph. Also, motivated by the Chinese Postman Problem, Boesch et al. introduced supereulerian graphs which contain spanning closed trails. In the spanning trail part of my dissertation, I proved some results based on supereulerian graphs and, a more general case, spanning trails.;Let alpha(G), alpha\u27(G), kappa( G) and kappa\u27(G) denote the independence number, the matching number, connectivity and edge connectivity of a graph G, respectively. First, we discuss the 3-edge-connected graphs with bounded edge-cuts of size 3, and prove that any 3-edge-connected graph with at most 11 edge cuts of size 3 is supereulerian, which improves Catlin\u27s result. Second, having the idea from Chvatal-Erdos Theorem which states that every graph G with kappa(G) ≥ alpha( G) is hamiltonian, we find families of finite graphs F 1 and F2 such that if a connected graph G satisfies kappa\u27(G) ≥ alpha(G) -- 1 (resp. kappa\u27(G) ≥ 3 and alpha\u27( G) ≤ 7), then G has a spanning closed trail if and only if G is not contractible to a member of F1 (resp. F2). Third, by solving a conjecture posed in [Discrete Math. 306 (2006) 87-98], we prove if G is essentially 4-edge-connected, then for any edge subset X0 ⊆ E(G) with |X0| ≤ 3 and any distinct edges e, e\u27 2 ∈ E(G), G has a spanning ( e, e\u27)-trail containing all edges in X0.;The results on spanning trees in my dissertation concern spanning tree packing and covering. We find a characterization of spanning tree packing and covering based on degree sequence. Let tau(G) be the maximum number of edge-disjoint spanning trees in G, a(G) be the minimum number of spanning trees whose union covers E(G). We prove that, given a graphic sequence d = (d1, d2···dn) (d1 ≥ d2 ≥···≥ dn) and integers k2 ≥ k1 \u3e 0, there exists a simple graph G with degree sequence d satisfying k 1 ≤ tau(G) ≤ a(G) ≤ k2 if and only if dn ≥ k1 and 2k1(n -- 1) ≤ Sigmani =1 di ≤ 2k2( n -- 1 |I| -- 1) + 2Sigma i∈I di, where I = {lcub}i : di \u3c k2{rcub}

Decomposing graphs of high minimum degree into 4-cycles

If a graph G decomposes into edge-disjoint 4-cycles, then each vertex of G has even degree and 4 divides the number of edges in G. It is shown that these obvious necessary conditions are also sufficient when G is any simple graph having minimum degree at least (3132+on(1))n, where n is the number of vertices in G. This improves the bound given by Gustavsson (PhD Thesis, University of Stockholm, 1991), who showed (as part of a more general result) sufficiency for simple graphs with minimum degree at least (1-10-94+on(1))n. On the other hand, it is known that for arbitrarily large values of n there exist simple graphs satisfying the obvious necessary conditions, having n vertices and minimum degree 35n-1, but having no decomposition into edge-disjoint 4-cycles. We also show that if G is a bipartite simple graph with n vertices in each part, then the obvious necessary conditions for G to decompose into 4-cycles are sufficient when G has minimum degree at least (3132+on(1))n

Direct numerical simulation of instabilities in parallel flow with spherical roughness elements

Results from a direct numerical simulation of laminar flow over a flat surface with spherical roughness elements using a spectral-element method are given. The numerical simulation approximates roughness as a cellular pattern of identical spheres protruding from a smooth wall. Periodic boundary conditions on the domain's horizontal faces simulate an infinite array of roughness elements extending in the streamwise and spanwise directions, which implies the parallel-flow assumption, and results in a closed domain. A body force, designed to yield the horizontal Blasius velocity in the absence of roughness, sustains the flow. Instabilities above a critical Reynolds number reveal negligible oscillations in the recirculation regions behind each sphere and in the free stream, high-amplitude oscillations in the layer directly above the spheres, and a mean profile with an inflection point near the sphere's crest. The inflection point yields an unstable layer above the roughness (where U''(y) is less than 0) and a stable region within the roughness (where U''(y) is greater than 0). Evidently, the instability begins when the low-momentum or wake region behind an element, being the region most affected by disturbances (purely numerical in this case), goes unstable and moves. In compressible flow with periodic boundaries, this motion sends disturbances to all regions of the domain. In the unstable layer just above the inflection point, the disturbances grow while being carried downstream with a propagation speed equal to the local mean velocity; they do not grow amid the low energy region near the roughness patch. The most amplified disturbance eventually arrives at the next roughness element downstream, perturbing its wake and inducing a global response at a frequency governed by the streamwise spacing between spheres and the mean velocity of the most amplified layer

Cycles and 1-unconditional matrices

We characterize the 1-unconditional subsequences of the canonical basis (e_rc) of elementary matrices in the Schatten-von-Neumann class S^p . The set I of couples (r,c) must be the set of edges of a bipartite graph without cycles of even length 4<=l<=p if p is an even integer, and without cycles at all if p is a positive real number that is not an even integer. In the latter case, I is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space S^p_I spanned by (e_rc)_{(r,c)\in I} in S^p .Comment: 29 pages. This new version computes explicitly certain unconditionality constants, shows how our results generalize Varopoulos' work on V-Sidon sets, investigates the metric unconditional approximation property in the same contex