The Guessing Number of Undirected Graphs

Christofides, Demetres; Markström, Klas

doi:10.37236/679

Cited by 21 publications

(69 citation statements)

References 12 publications

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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%

“…Christofides and Markström [7] showed that, for a perfect graph G and any s, gn(G, s) = n − α where α is the size of the largest independent set in G. For example, the complete graph K n is a perfect graph with α = 1, so an optimal protocol on K n , wins with probability 1/s. The 3-cycle and the even-cycle C 2k (for any positive integer k) are both perfect graphs with α(C 3 ) = 1 and α(C 2k ) = k so gn(C 3 , s) = 2 and gn(C 2k , s) = k ∀ k.…”

Section: Introductionmentioning

confidence: 99%

“…Henceforth, we shall consider only the cycle graphs C n for odd n ≥ 5. In [7], it is shown that gn(C 5 , 2) = 5, and the analysis in [3] shows that gn(C n , 2) = n − 1 2 , for odd n ≥ 7.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Backround Materials and Notationmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

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Guessing Numbers of Odd Cycles

Atkins

Rombach

Skerman³

2017

Electron. J. Combin.

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“…The q-strict guessing number is g[D, q] := log q fix + [D, q], while the q-guessing number is g(D, q) := log q fix + (D, q). The guessing number and the strict guessing number tend to the same limit as q tends to infinity [8]:…”

Section: Number Of Fixed Pointsmentioning

confidence: 96%

“…In particular, if D is an odd undirected cycle, i.e. D = C 2k+1 for k ≥ 2, then H(C 2k+1 ) = k + 1/2 [8].…”

Section: Number Of Fixed Pointsmentioning

confidence: 99%

On the influence of the interaction graph on a finite dynamical system

Gadouleau

2019

Nat Comput

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A finite dynamical system (FDS) is a system of multivariate functions over a finite alphabet, that is typically used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. As such, a major problem is to determine the effect of the interaction graph on the dynamics of the FDS. In this paper, we are interested in three main properties of an FDS: the number of images (the so-called rank), the number of periodic points (the so-called periodic rank) and the number of fixed points. In particular, we investigate the minimum, average, and maximum number of images (or periodic points, or fixed points) of FDSs with a prescribed interaction graph and a given alphabet size; thus yielding nine quantities to study. The paper is split into two parts. The first part considers the minimum rank, for which we derive the first meaningful results known so far. In particular, we show that the minimum rank decreases with the alphabet size, thus yielding the definition of an absolute minimum rank. We obtain lower and upper bounds on this absolute minimum rank, and we give classification results for graphs with very low (or highest) rank. The second part is a comprehensive survey of the results obtained on the nine quantities described above. We not only give a review of known results, but we also give a list of relevant open questions. *

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On the possible values of the entropy of undirected graphs

Gadouleau

2017

Journal of Graph Theory

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The entropy of a digraph is a fundamental measure which relates network coding, information theory, and fixed points of finite dynamical systems. In this paper, we focus on the entropy of undirected graphs. We prove that for any integer k the number of possible values of the entropy of an undirected graph up to k is finite. We also determine all the possible values for the entropy of an undirected graph up to the value of four.

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The Guessing Number of Undirected Graphs

Cited by 21 publications

References 12 publications

Guessing Numbers of Odd Cycles

Guessing Numbers of Odd Cycles

On the influence of the interaction graph on a finite dynamical system

On the possible values of the entropy of undirected graphs

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