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

Spanning trees with few branch vertices

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch vertices. Asymptotically, this is best possible and solves, in less general form, a problem of Flandrin, Kaiser, Ku\u{z}el, Li and Ryj\'a\u{c}ek, which was originally motivated by an optimization problem in the design of optical networks.Comment: 20 pages, 2 figures, to appear in SIAM J. of Discrete Mat

Partitioning random graphs into monochromatic components

Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every $r$-colored complete graph can be partitioned into at most $r-1$ monochromatic components; this is a strengthening of a conjecture of Lov\'asz (1975) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into $r$ monochromatic components is possible for sufficiently large $r$-colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if $p\ge \left(\frac{27\log n}{n}\right)^{1/3}$, then a.a.s. in every $2$-coloring of $G(n,p)$ there exists a partition into two monochromatic components, and for $r\geq 2$ if $p\ll \left(\frac{r\log n}{n}\right)^{1/r}$, then a.a.s. there exists an $r$-coloring of $G(n,p)$ such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gy\'arf\'as (1977) about large monochromatic components in $r$-colored complete graphs. We show that if $p=\frac{\omega(1)}{n}$, then a.a.s. in every $r$-coloring of $G(n,p)$ there exists a monochromatic component of order at least $(1-o(1))\frac{n}{r-1}$.Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics Volume 24, Issue 1 (2017) Paper #P1.1

Ore-degree threshold for the square of a Hamiltonian cycle

A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured that every graph with minimum degree at least 2n/3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's theorem by proving that every graph with $deg(u) + deg(v) \geq n$ for every $uv \notin E(G)$ contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type version of P\'osa's conjecture for graphs on $n\geq n_0$ vertices using the regularity--blow-up method; consequently the $n_0$ is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller $n_0$, we believe that our method of proof will be of independent interest.Comment: 24 pages, 1 figure. In addition to some fixed typos, this updated version contains a simplified "connecting lemma" in Section 3.