The average order of a connected induced subgraph of a graph and union‐intersection systems

Abstract

Because connectivity is such a basic concept in graph theory, extremal problems concerning the average order of the connected induced subgraphs of a graph have been of notable interest. A particularly resistant open problem is whether or not, for a connected graph of order , all of whose vertices have degree at least 3, this average is at least . It is shown in this paper that if is a connected, vertex transitive graph, then the average order of the connected induced subgraphs of is at least .The extremal graph theory problems mentioned above lead to a broader theory. The concept of a Union‐Intersection System (UIS)  is introduced, being a finite set of points and a set of subsets of called blocks satisfying the following simple property for all : if , then . To generalize results on the average order of a connected induced subgraph of a graph, it is conjectured that if a UIS is, in various senses, “connected and regular,” then the average size of a block is at least half the number of points. We prove that if a union‐intersection set system is regular, completely irreducible, and nonredundant, then the average size of a block is at least half the number of points.