5,377 research outputs found
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
Discrepancy and Signed Domination in Graphs and Hypergraphs
For a graph G, a signed domination function of G is a two-colouring of the
vertices of G with colours +1 and -1 such that the closed neighbourhood of
every vertex contains more +1's than -1's. This concept is closely related to
combinatorial discrepancy theory as shown by Fueredi and Mubayi [J. Combin.
Theory, Ser. B 76 (1999) 223-239]. The signed domination number of G is the
minimum of the sum of colours for all vertices, taken over all signed
domination functions of G. In this paper, we present new upper and lower bounds
for the signed domination number. These new bounds improve a number of known
results.Comment: 12 page
Approximating Bin Packing within O(log OPT * log log OPT) bins
For bin packing, the input consists of n items with sizes s_1,...,s_n in
[0,1] which have to be assigned to a minimum number of bins of size 1. The
seminal Karmarkar-Karp algorithm from '82 produces a solution with at most OPT
+ O(log^2 OPT) bins.
We provide the first improvement in now 3 decades and show that one can find
a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is
achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation
using the Entropy Method from discrepancy theory. The result is constructive
via algorithms of Bansal and Lovett-Meka
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