Guessing Numbers of Odd Cycles

Abstract

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].

Explicit Polyhedral Bounds on Network Coding Rate Regions via Entropy Function Region: Algorithms, Symmetry, and Computation

“…The upper bound obtained using Γ 5 , in this case, is 5 2 (see eg. [66]). The problem dimension is reduced from 25 to 5 by exploiting symmetry.…”

Guessing Numbers and Extremal Graph Theory

“…They found the exact guessing numbers of a class of graphs that contains the perfect graphs, namely the graphs whose independence number equals the clique cover number of their complements. The guessing numbers of undirected trianglefree graphs [5] and odd cycles [3] have also been studied, but very few other graphs have known guessing number.…”

Guessing Numbers and Extremal Graph Theory

“…They found the exact guessing numbers of a class of graphs that contains the perfect graphs, namely the graphs whose independence number equals the clique cover number of their complements. The guessing numbers of undirected triangle free graphs [5] and odd cycles [3] have also been studied, but very few other graphs have known guessing number.…”