The dichromatic number of a digraph

Neumann-Lara, Víctor

doi:10.1016/0095-8956(82)90046-6

Cited by 183 publications

(136 citation statements)

References 2 publications

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“…Now we show that the (logarithm of the) dichromatic number defined in [29] is an upper bound on the Dilworth rate.…”

Section: B Dichromatic Number and Upper Boundsmentioning

confidence: 99%

“…Though originally defined to give a general framework for some problems in extremal set theory (see [16], [17]), Sperner capacity also has its own information theoretic relevance, see [12], [30], [7]. The dichromatic number is a generalization of the chromatic number to directed graphs introduced in [29]. Using the above mentioned relations we determine our new parameter for some specific directed graphs.…”

Section: Introductionmentioning

confidence: 99%

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Dilworth Rate: A Generalization of Witsenhausen’s Zero-Error Rate for Directed Graphs

Simonyi

Tóth

2015

IEEE Trans. Inform. Theory

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Abstract-We investigate a communication setup where a source output is sent through a free noisy channel first and an additional codeword is sent through a noiseless but expensive channel later. With the help of the second message the decoder should be able to decide with zero-error whether its decoding of the first message was error-free. This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs. We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner capacity and dichromatic number.We also look at the natural variant of the above problem, where the decoder should decode the first message with zeroerror, not only decide whether its earlier decoding was correct. In this case the Witsenhausen rate of an appropriately defined undirected graph turns out to be the relevant parameter.

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“…Now we show that the (logarithm of the) dichromatic number defined in [29] is an upper bound on the Dilworth rate.…”

Section: B Dichromatic Number and Upper Boundsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dilworth Rate: A Generalization of Witsenhausen’s Zero-Error Rate for Directed Graphs

Simonyi

Tóth

2015

IEEE Trans. Inform. Theory

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“…Interested readers are suggested referring to the text books [25], [36] and papers [21], [26] for more details.…”

Section: A Graph Theorymentioning

confidence: 99%

“…Formally, the dichromatic number [21] of a digraph D, χ A (D), is the smallest cardinality |X | of a color set X , so that it is possible to assign a color from X to each vertex of D such that for every color c ∈ X the sub-digraph induced by the vertices colored with c is acyclic. Dichromatic number χ A generalizes the notion of chromatic number χ from graphs to digraphs.…”

Section: A Graph Theorymentioning

confidence: 99%

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Topological Interference Management With Decoded Message Passing

Caire

2018

IEEE Trans. Inform. Theory

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Abstract-The topological interference management (TIM) problem studies partially-connected interference networks with no channel state information except for the network topology (i.e., connectivity graph) at the transmitters. In this paper, we consider a similar problem in the uplink cellular networks, while message passing is enabled at the receivers (e.g., base stations), so that the decoded messages can be routed to other receivers via backhaul links to help further improve network performance. For this TIM problem with decoded message passing (TIM-MP), we model the interference pattern by conflict digraphs, connect orthogonal access to the acyclic set coloring on conflict digraphs, and show that one-to-one interference alignment boils down to orthogonal access because of message passing. With the aid of polyhedral combinatorics, we identify the structural properties of certain classes of network topologies where orthogonal access achieves the optimal degrees-of-freedom (DoF) region in the information-theoretic sense. The relation to the conventional index coding with simultaneous decoding is also investigated by formulating a generalized index coding problem with successive decoding as a result of decoded message passing. The properties of reducibility and criticality are also studied, by which we are able to prove the linear optimality of orthogonal access in terms of symmetric DoF for the networks up to four users with all possible network topologies (218 instances). Practical issues of the tradeoff between the overhead of message passing and the achievable symmetric DoF are also discussed, in the hope of facilitating efficient backhaul utilization.

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Sparse Graphs

and¹,

Toft²

1994

Graph Coloring Problems

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The dichromatic number of a digraph

Cited by 183 publications

References 2 publications

Dilworth Rate: A Generalization of Witsenhausen’s Zero-Error Rate for Directed Graphs

Dilworth Rate: A Generalization of Witsenhausen’s Zero-Error Rate for Directed Graphs

Topological Interference Management With Decoded Message Passing

Sparse Graphs

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