Deepak Bal

Rainbow matchings and Hamilton cycles in random graphs

Let HPn,m,k be drawn uniformly from all m-edge, k-uniform, k-partite hypergraphs where each part of the partition is a disjoint copy of [n]. We let HPn,m,k(κ) be an edge colored version, where we color each edge randomly from one of κ colors. We show that if κ=n and m=Knlogn where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in which every edge has a different color. We also show that if n is even and m=Knlogn where K is sufficiently large then w.h.p. there is a rainbow colored Hamilton cycle in Gn,m(n). Here Gn,m(n) denotes a random edge coloring of Gn,m with n colors. When n is odd, our proof requires m=ω(nlogn) for there to be a rainbow Hamilton cycle.

Packing Tight Hamilton Cycles in Uniform Hypergraphs

Consider a 3-uniform hypergraph <i>H</i> with <i>n</i> vertices. A tight Hamilton cycle <i>C</i> ⊂ <i>H</i> is a collection of <i>n</i> edges for which there is an ordering of the vertices <i>v</i>₁,..., <i>v</i>_<i>n</i> where every triple of consecutive vertices {<i>v</i>_<i>i</i>, <i>v</i>_<i>i</i>+1, <i>v</i>_<i>i</i>+2} is an edge of <i>C</i> (indices considered modulo <i>n</i>). We develop new techniques which show that under certain natural pseudo-random conditions, almost all edges of <i>H</i> can be covered by edge-disjoint tight Hamilton cycles, for <i>n</i> divisible by 4. Consequently, random 3-uniform hypergraphs can be almost completely packed with tight Hamilton cycles <b>whp</b>, for <i>n</i> divisible by 4 and <i>p</i> not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo-random digraphs with even numbers of vertices.

Lazy Cops and Robbers on Hypercubes

We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes.

A greedy algorithm for finding a large 2-matching on a random cubic graph

A 2-matching of a graph

On the maximum number of edges in a hypergraph with a unique perfect matching

G

Rainbow perfect matchings and Hamilton cycles in the random geometric graph

is a spanning subgraph with maximum degree two. The size of a 2-matching

Graph connectivity, partial words, and a theorem of Fine and Wilf

U

Partitioning Random Graphs into Monochromatic Components

is the number of edges in

Lazy Cops and Robbers played on random graphs and graphs on surfaces

U

The t-Tone Chromatic Number of Random Graphs

and this is at least