21,116 research outputs found
Nombre chromatique fractionnaire, degré maximum et maille
We prove new lower bounds on the independence ratio of graphs of maximum degree â â {3,4,5} and girth g â {6,âŠ,12}, notably 1/3 when (â,g)=(4,10) and 2/7 when (â,g)=(5,8). We establish a general upper bound on the fractional chromatic number of triangle-free graphs, which implies that deduced from the fractional version of Reed's bound for triangle-free graphs and improves it as soon as â â„ 17, matching the best asymptotic upper bound known for off-diagonal Ramsey numbers. In particular, the fractional chromatic number of a triangle-free graph of maximum degree â is less than 9.916 if â=17, less than 22.17 if â=50 and less than 249.06 if â=1000. Focusing on smaller values of â, we also demonstrate that every graph of girth at least 7 and maximum degree â has fractional chromatic number at most min (2â + 2^{k-3}+k)/k pour k â â. In particular, the fractional chromatic number of a graph of girth 7 and maximum degree â is at most (2â+9)/5 when â â [3,8], at most (â+7)/3 when â â [8,20], at most (2â+23)/7 when â â [20,48], and at most â/4+5 when â â [48,112]
Fractional Perfect b-Matching Polytopes. I: General Theory
The fractional perfect b-matching polytope of an undirected graph G is the
polytope of all assignments of nonnegative real numbers to the edges of G such
that the sum of the numbers over all edges incident to any vertex v is a
prescribed nonnegative number b_v. General theorems which provide conditions
for nonemptiness, give a formula for the dimension, and characterize the
vertices, edges and face lattices of such polytopes are obtained. Many of these
results are expressed in terms of certain spanning subgraphs of G which are
associated with subsets or elements of the polytope. For example, it is shown
that an element u of the fractional perfect b-matching polytope of G is a
vertex of the polytope if and only if each component of the graph of u either
is acyclic or else contains exactly one cycle with that cycle having odd
length, where the graph of u is defined to be the spanning subgraph of G whose
edges are those at which u is positive.Comment: 37 page
Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels
In this paper we study conditions which guarantee the existence of perfect
matchings and perfect fractional matchings in uniform hypergraphs. We reduce
this problem to an old conjecture by Erd\H{o}s on estimating the maximum number
of edges in a hypergraph when the (fractional) matching number is given, which
we are able to solve in some special cases using probabilistic techniques.
Based on these results, we obtain some general theorems on the minimum
-degree ensuring the existence of perfect (fractional) matchings. In
particular, we asymptotically determine the minimum vertex degree which
guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also
discuss an application to a problem of finding an optimal data allocation in a
distributed storage system
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
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