Analogues of the Shannon Capacity of a Graph

Abstract

Publisher Summary This chapter presents a discussion on analogues of the Shannon capacity of a graph. This study was stimulated by analogies between two graph-theoretic notions, which arise from apparently unrelated applied problems in information theory and operations research. The graph theoretic concepts are the capacity of a graph and the ultimate chromatic number. There are similarities between the concepts of capacity and ultimate chromatic number. They are related explicitly as a part of a general framework and several scattered results and methods are summarized and extended. The chapter introduces and discusses, in brief, a notion of degree of perfectness of a graph. The chapter also discusses the following: (1) product numbers, (2) ultimate and fractional numbers, (3) degrees of perfectness, (4) relations among the numbers, and (5) odd cycles. The vertex-set of a graph G is denoted by V ( G ), the edge-set by E ( G ). The independence number α ( G ) is the largest number of vertices in an independent set of G ; the clique number ω( G ) is the largest number of vertices in a clique (maximal complete subgraph) of G ; the chromatic number χ( G ) is the smallest number of independent sets of G whose union is V ( G ); the clique covering number θ ( G ) is the smallest number of cliques of G whose union is V ( G ).