On the Shannon Capacity of Triangular Graphs

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Authors

ASHIK Mathew Kizhakkepallathu PATRIC R. J. Östergard POPA Alexandru

Year of publication 2013
Type Article in Periodical
Magazine / Source Electronic Journal of Combinatorics
MU Faculty or unit

Faculty of Informatics

Citation
Web http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i2p27
Field Information theory
Keywords cube packing; Shannon capacity; tabu search; zero-error capacity
Description The Shannon capacity of a graph $G$ is defined as $c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of $G$. The Shannon capacity of the Kneser graph $\kg{n}{r}$ was determined by Lov\'{a}sz in 1979, but little is known about the Shannon capacity of the complement of that graph when $r$ does not divide $n$. The complement of the Kneser graph, $\kgc{n}{2}$, has the $n$-cycle $C_n$ as an induced subgraph, whereby $c(\kgc{n}{2}) \geq c(C_n)$, and these two families of graphs are closely related in the current context as both can be considered via geometric packings of the discrete $d$-dimensional torus of width $n$ using two types of $d$-dimensional cubes of width $2$. Bounds on $c(\kgc{n}{2})$ obtained in this work include $c(\kgc{7}{2}) \geq \sqrt[3]{35} \approx 3.271$, $c(\kgc{13}{2}) \geq \sqrt[3]{248} \approx 6.283$, $c(\kgc{15}{2}) \geq \sqrt[4]{2802} \approx 7.276$, and $c(\kgc{21}{2}) \geq \sqrt[4]{11441} \approx 10.342$.
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