Graph Guessing Games and non-Shannon Information Inequalities

Baber, Rahil; Christofides, Demetres; Dang, Anh N.; Riis, Søren; Vaughan, Emil R.

doi:10.48550/arxiv.1410.8349

Cited by 2 publications

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“…This also gives counterexamples to their conjecture about the optimal guessing strategy based on fractional clique cover in [8]. (The first counterexample to Christofides and Markström's conjecture was illustrated in [2] but the graph is not triangle-free. )…”

Section: Introductionmentioning

confidence: 92%

“…This leaves 16 blocks that have empty intersection with B. From (b) we know that any two blocks must intersect in zero or two points, this makes the 16 points and 16 blocks a symmetric balanced incomplete block design (BIBD) (16,6,2). It follows from the property of symmetric BIBD that any two blocks intersect in 2 points.…”

Section: Now We Fix One Point I In [N]mentioning

confidence: 99%

“…Our main results appear in Section 5. Sections 2, 3, 4 already appeared in [11], [8], and [2] but we reproduce them here in order to make this paper self-contained.…”

Section: Introductionmentioning

confidence: 99%

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Guessing Games on Triangle-Free Graphs

Cameron¹,

Dang²,

Riis³

2016

Electron. J. Combin.

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The guessing game introduced by Riis[17] is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markström [8] introduced a method to bound the value of the guessing number from below using the fractional clique cover number κ f (G). In particular they showed gn(G) ≥ |V (G)| − κ f (G). Moreover, it is pointed out that equality holds in this bound if the underlying undirected graph G falls into one of the following categories: perfect graphs, cycle graphs or their complement. In this paper, we show that there are triangle-free graphs that have guessing numbers which do not meet the fractional clique cover bound. In particular, the famous triangle-free Higman-Sims graph has guessing number at least 77 and at most 78, while the bound given by fractional clique cover is 50.Remark 5.10. For a more extensive list of computations of ranks of matrices A + kI over F q for q = 2, 3, 5, 7 see EBasis.zip at https://www.eecs.qmul.ac.uk/~smriis/.Problem 5.11. Find the exact value of the guessing number of the strongly regular graphs considered here.

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Section: Introductionmentioning

confidence: 92%

Section: Now We Fix One Point I In [N]mentioning

confidence: 99%

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Guessing Games on Triangle-Free Graphs

Cameron¹,

Dang²,

Riis³

2016

Electron. J. Combin.

Self Cite

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“…In the rest of this section, we present a few known results on the guessing number, define some useful random variables on the cycle graph and a notion of entropy, all of which will be used extensively in our proofs. When possible, we are consistent with the definitions and notations given in [5,6,7,2,13,14]. We start with a small, useful result that shows, intuitively, that we are allowed to "forget" some colours.…”

Section: Backround Materials and Notationmentioning

confidence: 85%

“…Computing the guessing number (Definition 1.2) of a graph G, can be equivalent to determining whether the multiple unicast coding problem [9] is solvable on a network related to G. The guessing number of a graph, G, is also studied for its relation to the information defect of G and index coding with side information [1,11]. Exact guessing numbers are known only for a small number specific classes of graphs, such as perfect graphs, or small cases of non-perfect graphs [2,5,6,15]. In particular, the guessing number of odd cycles, which is the focus of this paper, was not known, except for small cases [7,3].…”

Section: Introductionmentioning

confidence: 99%

Guessing Numbers of Odd Cycles

Atkins

Rombach

Skerman³

2017

Electron. J. Combin.

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For a given number of colours, s, the guessing number of a graph is the base s logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the n-vertex cycle graph Cn is n/2. It is known that the guessing number equals n/2 whenever n is even or s is a perfect square [7]. We show that, for any given integer s ≥ 2, if a is the largest factor of s less than or equal to √ s, for sufficiently large odd n, the guessing number of Cn with s colours is (n − 1)/2 + log s (a). This answers a question posed by Christofides and Markström in 2011 [7]. We also present an explicit protocol which achieves this bound for every n. Linking this to index coding with side information, we deduce that the information defect of Cn with s colours is (n + 1)/2 − log s (a) for sufficiently large odd n. Our results are a generalisation of the s = 2 case which was proven in [3].

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Graph Guessing Games and non-Shannon Information Inequalities

Cited by 2 publications

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Guessing Games on Triangle-Free Graphs

Guessing Games on Triangle-Free Graphs

Guessing Numbers of Odd Cycles

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