3,121 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
The min-max edge q-coloring problem
In this paper we introduce and study a new problem named \emph{min-max edge
-coloring} which is motivated by applications in wireless mesh networks. The
input of the problem consists of an undirected graph and an integer . The
goal is to color the edges of the graph with as many colors as possible such
that: (a) any vertex is incident to at most different colors, and (b) the
maximum size of a color group (i.e. set of edges identically colored) is
minimized. We show the following results: 1. Min-max edge -coloring is
NP-hard, for any . 2. A polynomial time exact algorithm for min-max
edge -coloring on trees. 3. Exact formulas of the optimal solution for
cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial
lower bound of the optimal solution with respect to the average degree of the
graph. 5. An approximation algorithm for planar graphs.Comment: 16 pages, 5 figure
All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs
For an undirected -vertex graph with non-negative edge-weights, we
consider the following type of query: given two vertices and in ,
what is the weight of a minimum -cut in ? We solve this problem in
preprocessing time for graphs of bounded genus, giving the first
sub-quadratic time algorithm for this class of graphs. Our result also improves
by a logarithmic factor a previous algorithm by Borradaile, Sankowski and
Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm
constructs a Gomory-Hu tree for the given graph, providing a data structure
with space that can answer minimum-cut queries in constant time. The
dependence on the genus of the input graph in our preprocessing time is
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