Graph Invariants and Large Cycles - a Catalog of Pure Links

Abstract

Graph invariants provide a powerful analytical tool for investigation of abstract structures of graphs. They, combined in convenient relations, carry global and general information about a graph and its various substructures such as cycle structures, factors, matchings, colorings, coverings, and so on, whose discovery is the primary problem of graph theory. The major goal of this paper is to catalogue all pure relations between basic invariants of a graph and its large cycle structures, namely Hamilton, longest and dominating cycles and some their generalizations. Basic graph invariants and pure relations allow to focus on results having no forerunners. These simplest kind of "ancestors" form a source from which nearly all possible hamiltonian results can be developed further by various additional new ideas, generalizations, extensions, restrictions and structural limitations, as well as helping researchers to make clear and simple imagination about "complicated" developmental mechanisms in the area.