183,446 research outputs found
Spanning Eulerian subgraphs and Catlin’s reduced graphs
A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph HR of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if G satisfies one of the following:
(i) d(u) + d(v) \u3e 2(n/15 − 1) for any uv 6∈ E(G) and n is large;
(ii) the size of a maximum matching in G is at most 6;
(iii) the independence number of G is at most 5.
These are improvements of prior results in [16], [18], [24] and [25]
Many-Access Channels: The Gaussian Case with Random User Activities
Classical multiuser information theory studies the fundamental limits of
models with a fixed (often small) number of users as the coding blocklength
goes to infinity. This work proposes a new paradigm, referred to as many-user
information theory, where the number of users is allowed to grow with the
blocklength. This paradigm is motivated by emerging systems with a massive
number of users in an area, such as machine-to-machine communication systems
and sensor networks. The focus of the current paper is the many-access channel
model, which consists of a single receiver and many transmitters, whose number
increases unboundedly with the blocklength. Moreover, an unknown subset of
transmitters may transmit in a given block and need to be identified. A new
notion of capacity is introduced and characterized for the Gaussian many-access
channel with random user activities. The capacity can be achieved by first
detecting the set of active users and then decoding their messages.Comment: 5 pages, 2 figures, to appear in Proceedings of ISIT 201
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