54,562 research outputs found
Flows and bisections in cubic graphs
A -weak bisection of a cubic graph is a partition of the vertex-set of
into two parts and of equal size, such that each connected
component of the subgraph of induced by () is a tree of at
most vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph with a circular nowhere-zero -flow has a -weak bisection. In this paper we study problems related to the
existence of -weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio
Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs
In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero
5-flow. Let be the minimum number of odd cycles in a 2-factor of a
bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to
cubic graphs with . We show that if a cubic graph has no
edge cut with fewer than edges that separates two odd
cycles of a minimum 2-factor of , then has a nowhere-zero 5-flow. This
implies that if a cubic graph is cyclically -edge connected and , then has a nowhere-zero 5-flow
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Complete algebraic vector fields on affine surfaces
Let \AAutH (X) be the subgroup of the group \AutH (X) of holomorphic
automorphisms of a normal affine algebraic surface generated by elements of
flows associated with complete algebraic vector fields. Our main result is a
classification of all normal affine algebraic surfaces quasi-homogeneous
under \AAutH (X) in terms of the dual graphs of the boundaries \bX \setminus
X of their SNC-completions \bX.Comment: 44 page
Space Shuffle: A Scalable, Flexible, and High-Bandwidth Data Center Network
Data center applications require the network to be scalable and
bandwidth-rich. Current data center network architectures often use rigid
topologies to increase network bandwidth. A major limitation is that they can
hardly support incremental network growth. Recent work proposes to use random
interconnects to provide growth flexibility. However routing on a random
topology suffers from control and data plane scalability problems, because
routing decisions require global information and forwarding state cannot be
aggregated. In this paper we design a novel flexible data center network
architecture, Space Shuffle (S2), which applies greedy routing on multiple ring
spaces to achieve high-throughput, scalability, and flexibility. The proposed
greedy routing protocol of S2 effectively exploits the path diversity of
densely connected topologies and enables key-based routing. Extensive
experimental studies show that S2 provides high bisectional bandwidth and
throughput, near-optimal routing path lengths, extremely small forwarding
state, fairness among concurrent data flows, and resiliency to network
failures
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