Gallai theorems for graphs, hypergraphs, and set systems

Abstract

Publisher Summary In 1959 Gallai presented his now classical theorem, involving the vertex covering number α 0 , the vertex independence number β 0 , the edge covering number α 1 , and the maximum matching (or edge independence) number β 1 . This chapter attempts to collect and unify results of Gallai Theorems. It presents two general theorems that encompass nearly all of the existing Gallai theorems. The first theorem is based on hereditary properties of set systems, while the second is based on partitions of vertices into subgraphs having treelike properties. It also present a variety of new Gallai theorems (one of which is not a corollary of either of the two generalizations), as well as a number of other new results. The chapter proves a general theorem concerning hereditary properties of sets, from which one can obtain as corollaries a variety of Gallai theorems, including the original Gallai Theorem.