190 research outputs found
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 , every connected graph on vertices with minimum
degree at least contains a spanning tree having at most
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 -colored
complete graph can be partitioned into at most 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 monochromatic
components is possible for sufficiently large -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 , then a.a.s. in every -coloring of there exists
a partition into two monochromatic components, and for if , then a.a.s. there exists an -coloring
of 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 -colored complete
graphs. We show that if , then a.a.s. in every
-coloring of there exists a monochromatic component of order at
least .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 for every contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type
version of P\'osa's conjecture for graphs on vertices using the
regularity--blow-up method; consequently the 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
, 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.
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