Béla Csaba

Local resilience of almost spanning trees in random graphs

We prove that for fixed integer D and positive reals α and γ, there exists a constant C0 such that for all p satisfying p(n) ≥ C0/n, the random graph G(n,p) asymptotically almost surely contains a copy of every tree with maximum degree at most D and at most (1 - α)n vertices, even after we delete a (1/2 - γ)-fraction of the edges incident to each vertex. The proof uses Szemeredis regularity lemma for sparse graphs and a bipartite variant of the theorem of Friedman and Pippenger on embedding bounded degree trees into expanding graphs.

Proof of a Conjecture of Bollobás and Eldridge for Graphs of Maximum Degree Three

Let G1 and G2 be simple graphs on n vertices. If there are edge-disjoint copies of G1 and G2 in Kn, then we say there is a packing of G1 and G2. A conjecture of Bollobás and Eldridge [5] asserts that if (Δ(G1)+1) (Δ(G2)+1) ≤ n + 1 then there is a packing of G1 and G2. We prove this conjecture when Δ(G1) = 3, for sufficiently large n.

On the Bollobás–Eldridge Conjecture for Bipartite Graphs

Let G be a simple graph on n vertices. A conjecture of Bollobas and Eldridge [5] asserts that if δ(G) ≥ kn−1 k+1 then G contains any n vertex graph H with ∆(H) = k. We prove a strengthened version of this conjecture for bipartite, bounded degree H, for sufficiently large n. This is the first result on this conjecture for expander graphs of arbitrary (but bounded) degree. An important tool for the proof is a new version of the Blow-up Lemma.

Tight bounds for embedding bounded degree trees

Let T be a tree on n vertices with constant maximum degree K. Let G be a graph on n vertices having minimum degree δ(G) ≥ n/2 + ck log n, where CK is a constant. If n is sufficiently large then T ⊂ G. We also show that the bound on the minimum degree of G is tight.

Optimal Random Matchings on Trees and Applications

In this paper we will consider tight upper and lower bounds on the weight of the optimal matching for random point sets distributed among the leaves of a tree, as a function of its cardinality. Specifically, given two nsets of points R= {r 1 ,...,r n } and B= {b 1 ,...,b n } distributed uniformly and randomly on the mleaves of i¾?-Hierarchically Separated Trees with branching factor bsuch that each of its leaves is at depth i¾?, we will prove that the expected weight of optimal matching between Rand Bis

Optimal random matchings, tours, and spanning trees in hierarchically separated trees

\Theta(\sqrt{nb}\sum_{k=1}^h(\sqrt{b}\l)^k)

Large bounded degree trees in expanding graphs

, for h= min (i¾?,log b n). Using a simple embedding algorithm from i¾?dto HSTs, we are able to reproduce the results concerning the expected optimal transportation cost in [0,1]d, except for d= 2. We also show that giving random weights to the points does not affect the expected matching weight by more than a constant factor. Finally, we prove upper bounds on several sets for which showing reasonable matching results would previously have been intractable, e.g., the Cantor set, and various fractals.

Approximability of dense and sparse instances of minimum 2-connectivity, TSP and path problems

We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size n distributed among the leaves of a balanced hierarchically separated tree. We consider monochromatic and bichromatic versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.