691 research outputs found
Ramsey numbers of cubes versus cliques
The cube graph Q_n is the skeleton of the n-dimensional cube. It is an
n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N
such that every graph of order N contains the cube graph Q_n or an independent
set of order s. Burr and Erdos in 1983 asked whether the simple lower bound
r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large.
We make progress on this problem, obtaining the first upper bound which is
within a constant factor of the lower bound.Comment: 26 page
Bipartite induced density in triangle-free graphs
We prove that any triangle-free graph on vertices with minimum degree at
least contains a bipartite induced subgraph of minimum degree at least
. This is sharp up to a logarithmic factor in . Relatedly, we show
that the fractional chromatic number of any such triangle-free graph is at most
the minimum of and as . This is
sharp up to constant factors. Similarly, we show that the list chromatic number
of any such triangle-free graph is at most as
.
Relatedly, we also make two conjectures. First, any triangle-free graph on
vertices has fractional chromatic number at most
as . Second, any triangle-free
graph on vertices has list chromatic number at most as
.Comment: 20 pages; in v2 added note of concurrent work and one reference; in
v3 added more notes of ensuing work and a result towards one of the
conjectures (for list colouring
Ramsey numbers and the size of graphs
For two graph H and G, the Ramsey number r(H, G) is the smallest positive
integer n such that every red-blue edge coloring of the complete graph K_n on n
vertices contains either a red copy of H or a blue copy of G. Motivated by
questions of Erdos and Harary, in this note we study how the Ramsey number
r(K_s, G) depends on the size of the graph G. For s \geq 3, we prove that for
every G with m edges, r(K_s,G) \geq c (m/\log m)^{\frac{s+1}{s+3}} for some
positive constant c depending only on s. This lower bound improves an earlier
result of Erdos, Faudree, Rousseau, and Schelp, and is tight up to a
polylogarithmic factor when s=3. We also study the maximum value of r(K_s,G) as
a function of m
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