Compactness Properties of Weighted Summation Operators on Trees

We investigate compactness properties of weighted summation operators $V_{\alpha,\sigma}$ as mapping from $\ell_1(T)$ into $\ell_q(T)$ for some $q\in (1,\infty)$. Those operators are defined by $$(V_{\alpha,\sigma} x)(t) :=\alpha(t)\sum_{s\succeq t}\sigma(s) x(s)\,,\quad t\in T\;,$$ where $T$ is a tree with induced partial order $t \preceq s$ (or $s \succeq t$) for $t,s\in T$. Here $\alpha$ and $\sigma$ are given weights on $T$. We introduce a metric $d$ on $T$ such that compactness properties of $(T,d)$ imply two--sided estimates for $e_n(V_{\alpha,\sigma})$, the (dyadic) entropy numbers of $V_{\alpha,\sigma}$. The results are applied for concrete trees as e.g. moderate increasing, biased or binary trees and for weights with $\alpha(t)\sigma(t)$ decreasing either polynomially or exponentially. We also give some probabilistic applications for Gaussian summation schemes on trees

On the tractability of some natural packing, covering and partitioning problems

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mathrm{B}$? \textbf{Covering problem:} can we cover $E$ with an object of type $\mathrm{A}$, and an object of type $\mathrm{B}$? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an $s$-$t$ path $P$ and an $s'$-$t'$ path $P'$ that are edge-disjoint. However, many others were not, for example can we find an $s$-$t$ path $P\subseteq E$ and a spanning tree $T\subseteq E$ that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

On Generalizations of Supereulerian Graphs

A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let $\kappa\u27(G)$ and $\delta(G)$ be the edge-connectivity and the minimum degree of a graph $G$, respectively. For integers $s \ge 0$ and $t \ge 0$, a graph $G$ is $(s,t)$-supereulerian if for any disjoint edge sets $X, Y \subseteq E(G)$ with $|X|\le s$ and $|Y|\le t$, $G$ has a spanning closed trail that contains $X$ and avoids $Y$. This dissertation is devoted to providing some results on $(s,t)$-supereulerian graphs and supereulerian hypergraphs. In Chapter 2, we determine the value of the smallest integer $j(s,t)$ such that every $j(s,t)$-edge-connected graph is $(s,t)$-supereulerian as follows: j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right. As applications, we characterize $(s,t)$-supereulerian graphs when $t \ge 3$ in terms of edge-connectivities, and show that when $t \ge 3$, $(s,t)$-supereulerianicity is polynomially determinable. In Chapter 3, for a subset $Y \subseteq E(G)$ with $|Y|\le \kappa\u27(G)-1$, a necessary and sufficient condition for $G-Y$ to be a contractible configuration for supereulerianicity is obtained. We also characterize the $(s,t)$-supereulerianicity of $G$ when $s+t\le \kappa\u27(G)$. These results are applied to show that if $G$ is $(s,t)$-supereulerian with $\kappa\u27(G)=\delta(G)\ge 3$, then for any permutation $\alpha$ on the vertex set $V(G)$, the permutation graph $\alpha(G)$ is $(s,t)$-supereulerian if and only if $s+t\le \kappa\u27(G)$. For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Let $i_{s,t}(G)$ and $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $(s,t)$-supereulerian and $s$-Hamiltonian, respectively. In Chapter 4, for a simple graph $G$, we establish upper bounds for $i_{s,t}(G)$ and $h_s(G)$. Specifically, the upper bound for the $s$-Hamiltonian index $h_s(G)$ sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785]. Harary and Nash-Williams in 1968 proved that the line graph of a graph $G$ is Hamiltonian if and only if $G$ has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity. Applying the adjacency matrix of a hypergraph $H$ defined by Rodr\\u27iguez in 2002, let $\lambda_2(H)$ be the second largest adjacency eigenvalue of $H$. In Chapter 6, we prove that for an integer $k$ and a $r$-uniform hypergraph $H$ of order $n$ with $r\ge 4$ even, the minimum degree $\delta\ge k\ge 2$ and $k\neq r+2$, if $\lambda_2(H)\le (r-1)\delta-\frac{r^2(k-1)n}{4(r+1)(n-r-1)}$, then $H$ is $k$-edge-connected. %$\kappa\u27(H)\ge k$. Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The $(s,t)$-supereulerianicity of hypergraphs is another interesting topic to be investigated in the future

Decomposition of Levy trees along their diameter

We study the diameter of L{\'e}vy trees that are random compact metric spaces obtained as the scaling limits of Galton-Watson trees. L{\'e}vy trees have been introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum Random Tree (1991) that corresponds to the Brownian case. We first characterize the law of the diameter of L{\'e}vy trees and we prove that it is realized by a unique pair of points. We prove that the law of L{\'e}vy trees conditioned to have a fixed diameter r $\in$ (0, $\infty$) is obtained by glueing at their respective roots two independent size-biased L{\'e}vy trees conditioned to have height r/2 and then by uniformly re-rooting the resulting tree; we also describe by a Poisson point measure the law of the subtrees that are grafted on the diameter. As an application of this decomposition of L{\'e}vy trees according to their diameter, we characterize the joint law of the height and the diameter of stable L{\'e}vy trees conditioned by their total mass; we also provide asymptotic expansions of the law of the height and of the diameter of such normalised stable trees, which generalises the identity due to Szekeres (1983) in the Brownian case