Spanning Trees of Bounded Degree Graphs

We consider lower bounds on the number of spanning trees of connected graphs with degree bounded by $d$. The question is of interest because such bounds may improve the analysis of the improvement produced by memorisation in the runtime of exponential algorithms. The value of interest is the constant $\beta_d$ such that all connected graphs with degree bounded by $d$ have at least $\beta_d^\mu$ spanning trees where $\mu$ is the cyclomatic number or excess of the graph, namely $m-n+1$. We conjecture that $\beta_d$ is achieved by the complete graph $K_{d+1}$ but we have not proved this for any $d$ greater than 3. We give weaker lower bounds on $\beta_d$ for $d\le 11$

Proximity Drawings of High-Degree Trees

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set

Minimum vertex degree conditions for loose spanning trees in 3-graphs

In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all $\gamma$ and $\Delta$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5/9 + \gamma)\binom{n}{2}$ contains every loose spanning tree $T$ with maximum vertex degree $\Delta$. This bound is asymptotically tight, since some loose trees contain perfect matchings.Comment: 18 pages, 1 figur

Sharp threshold for embedding combs and other spanning trees in random graphs

When $k|n$, the tree $\mathrm{Comb}_{n,k}$ consists of a path containing $n/k$ vertices, each of whose vertices has a disjoint path length $k-1$ beginning at it. We show that, for any $k=k(n)$ and $\epsilon>0$, the binomial random graph $\mathcal{G}(n,(1+\epsilon)\log n/ n)$ almost surely contains $\mathrm{Comb}_{n,k}$ as a subgraph. This improves a recent result of Kahn, Lubetzky and Wormald. We prove a similar statement for a more general class of trees containing both these combs and all bounded degree spanning trees which have at least $\epsilon n/ \log^9n$ disjoint bare paths length $\lceil\log^9 n\rceil$. We also give an efficient method for finding large expander subgraphs in a binomial random graph. This allows us to improve a result on almost spanning trees by Balogh, Csaba, Pei and Samotij.Comment: 20 page

A push–relabel approximation algorithm for approximating the minimum-degree MST problem and its generalization to matroids

AbstractIn the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to find a minimum spanning tree (MST) T, such that the maximum degree of T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST of degree at most 2Δopt (G)+o(Δopt (G)), where Δopt (G) denotes the degree of the optimal tree. This result improves on a previous result of Fischer [T. Fischer, Optimizing the degree of minimum weight spanning trees. Technical Report 14853, Dept. of Computer Science, Cornell University, Ithaca, NY, 1993] that finds an MST of degree at most bΔopt (G)+logbn, for any b>1.The MDMST problem is a special case of the following problem: given a k-ary hypergraph G=(V,E) and weighted matroid M with E as its ground set, find a minimum-cost basis (MCB) T of M such that the degree of T in G is as small as possible. Our algorithm immediately generalizes to this problem, finding an MCB of degree at most k2Δopt (G,M)+O(kkΔopt (G,M)).We use the push–relabel framework developed by Goldberg [A. V. Goldberg, A new max-flow algorithm, Technical Report MIT/LCS/TM-291, Massachusetts Institute of Technology, 1985 (Technical Report)] for the maximum-flow problem. To our knowledge, this is the first use of the push–relabel technique in an approximation algorithm for an NP-hard problem.The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms for this problem by Könemann and Ravi [J. Könemann, R. Ravi, A matter of degree: Improved approximation algorithms for degree-bounded minimum spanning trees, SIAM Journal on Computing 31(6) (2002) 1783–1793; J. Könemann, R. Ravi, Primal-dual meets local search: Approximating MST’s with nonuniform degree bounds, in: Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing, 2003, pp. 389–395] and by Chaudhuri et al. [K. Chaudhuri, S. Rao, S. Riesenfeld, K. Talwar, What would Edmonds do? Augmenting paths and witnesses for bounded degree MSTs, in: Proceedings of APPROX/RANDOM, 2005, pp. 26–39] incur a near-logarithmic additive error in the degree. We give the first BDMST algorithm that approximates both the degree and the cost to within a constant factor of the optimum. These results generalize to the case of nonuniform degree bounds

Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal

In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G=(V,E) with a degree upper bound Bv on each vertex v∈V, and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this paper, we present a polynomial time algorithm which returns a spanning tree T of cost at most OPT and dT(v)≤Bv+1 for all v, where dT(v) denotes the degree of v in T. This generalizes a result of Fürer and Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degree upper bound Bv, and returns a spanning tree with cost at most OPT and Av−1≤dT(v) ≤ Bv+1 for all v ∈ V. This is essentially the best possible. The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms

A Local Algorithm for the Sparse Spanning Graph Problem

Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries. Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees, cannot be constructed efficiently in this model. Therefore, we settle for a spanning subgraph containing at most $(1+\varepsilon)n$ edges (where $n$ is the number of vertices and $\varepsilon$ is a given approximation/sparsity parameter). We achieve query complexity of $\tilde{O}(poly(\Delta/\varepsilon)n^{2/3})$, ($\tilde{O}$-notation hides polylogarithmic factors in $n$). where $\Delta$ is the maximum degree of the input graph. Our algorithm is the first to do so on arbitrary bounded degree graphs. Moreover, we achieve the additional property that our algorithm outputs a spanner, i.e., distances are approximately preserved. With high probability, for each deleted edge there is a path of $O(poly(\Delta/\varepsilon)\log^2 n)$ hops in the output that connects its endpoints

The bandwidth theorem in sparse graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175–205, 2009] states that any n-vertex graph G with minimum degree [Formula Presented] contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for p ≫[Formula Presented] asymptotically almost surely each spanning subgraph G of G(n, p) with minimum degree [Formula Presented] pn contains all n-vertex k-colourable graphs H with maximum degree ∆, bandwidth o(n), and at least Cp−2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p ≫[Formula Presented]

A Centralized Local Algorithm for the Sparse Spanning Graph Problem

Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries. Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees, cannot be constructed efficiently in this model. Therefore, we settle for a spanning subgraph containing at most (1+epsilon)n edges (where n is the number of vertices and epsilon is a given approximation/sparsity parameter). We achieve a query complexity of O~(poly(Delta/epsilon)n^{2/3}), where Delta is the maximum degree of the input graph. Our algorithm is the first to do so on arbitrary bounded degree graphs. Moreover, we achieve the additional property that our algorithm outputs a spanning subgraph of bounded stretch i.e., distances are approximately preserved. With high probability, for each deleted edge there is a path of O(log n * (Delta+log n)/epsilon) hops in the output that connects its endpoints