44,392 research outputs found
Fractional total colourings of graphs of high girth
Reed conjectured that for every epsilon>0 and Delta there exists g such that
the fractional total chromatic number of a graph with maximum degree Delta and
girth at least g is at most Delta+1+epsilon. We prove the conjecture for
Delta=3 and for even Delta>=4 in the following stronger form: For each of these
values of Delta, there exists g such that the fractional total chromatic number
of any graph with maximum degree Delta and girth at least g is equal to
Delta+1
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly
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
Locally Optimal Load Balancing
This work studies distributed algorithms for locally optimal load-balancing:
We are given a graph of maximum degree , and each node has up to
units of load. The task is to distribute the load more evenly so that the loads
of adjacent nodes differ by at most .
If the graph is a path (), it is easy to solve the fractional
version of the problem in communication rounds, independently of the
number of nodes. We show that this is tight, and we show that it is possible to
solve also the discrete version of the problem in rounds in paths.
For the general case (), we show that fractional load balancing
can be solved in rounds and discrete load
balancing in rounds for some function , independently of the
number of nodes.Comment: 19 pages, 11 figure
Some results on chromatic number as a function of triangle count
A variety of powerful extremal results have been shown for the chromatic
number of triangle-free graphs. Three noteworthy bounds are in terms of the
number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994),
and Johansson. There have been comparatively fewer works extending these types
of bounds to graphs with a small number of triangles. One noteworthy exception
is a result of Alon et. al (1999) bounding the chromatic number for graphs with
low degree and few triangles per vertex; this bound is nearly the same as for
triangle-free graphs. This type of parametrization is much less rigid, and has
appeared in dozens of combinatorial constructions.
In this paper, we show a similar type of result for as a function
of the number of vertices , the number of edges , as well as the triangle
count (both local and global measures). Our results smoothly interpolate
between the generic bounds true for all graphs and bounds for triangle-free
graphs. Our results are tight for most of these cases; we show how an open
problem regarding fractional chromatic number and degeneracy in triangle-free
graphs can resolve the small remaining gap in our bounds
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