Tiling in bipartite graphs with asymmetric minimum degrees

The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladk\'y and Schacht, and Czygrinow and DeBiasio determined the optimal minimum degree condition for a balanced bipartite graph on 2m(s+t) vertices to contain m vertex disjoint copies of K_{s,t} for fixed positive integers s<t. For a balanced bipartite graph G[U,V], let \delta_U be the minimum degree over all vertices in U and \delta_V be the minimum degree over all vertices in V. We consider the problem of determining the optimal value of \delta_U+\delta_V which guarantees that G can be tiled with K_{s,s}. We show that the optimal value depends on D:=|\delta_V-\delta_U|. When D is small, we show that \delta_U+\delta_V\geq n+3s-5 is best possible. As D becomes larger, we show that \delta_U+\delta_V can be made smaller, but no smaller than n+2s-2s^{1/2}. However, when D=n-C for some constant C, we show that there exist graphs with \delta_U+\delta_V\geq n+s^{s^{1/3}} which cannot be tiled with K_{s,s}.Comment: 34 pages, 4 figures. This is the unabridged version of the paper, containing the full proof of Theorem 1.7. The case when |\delta_U-\delta_V| is small and s>2 involves a lengthy case analysis, spanning pages 20-32; this section is not included in the "journal version

Pebbling in Dense Graphs

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n>=9 vertices having minimum degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336 vertices in each part having minimum degree at least floor(m/2)+1. Both bounds are best possible. In addition, we prove that the pebbling threshold of graphs with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when d is proportional to n.Comment: 10 page

Distributed Approximations of f-Matchings and b-Matchings in Graphs of Sub-Logarithmic Expansion

We give a distributed algorithm which given ? > 0 finds a (1-?)-factor approximation of a maximum f-matching in graphs G = (V,E) of sub-logarithmic expansion. Using a similar approach we also give a distributed approximation of a maximum b-matching in the same class of graphs provided the function b: V ? ?^+ is L-Lipschitz for some constant L. Both algorithms run in O(log^* n) rounds in the LOCAL model, which is optimal