Maximum versus minimum invariants for graphs

Abstract

Two examples in graph theory of the concept of the maximum and minimum pair of invariants resulting from a graphical parameter are ( I ) the chromatic number and its maximum version, the achromatic number, and (2) the genus and the maximum genus. The primary purpose of this expository survey is to propose the generalization of this idea to each invariant p of a maximum or minimum nature. As illustrations, we consider for p the number of independent points or lines, the point arboricity, the connectivity, the rectilinear crossing number, the diameter of a spanning tree, and the conventional extremal number with respect to a forbidden subgraph. An additional objective is to describe the idea of an "interpolating invariant" which takes on, in a precise sense, all values between its minimum and its maximum, and to give examples.