On Orientations, Connectivity and Odd-Vertex-Pairings in Finite Graphs

Source abstract

The integer part of a non-negative real number p will be denoted by [p] . For any integer n, n* will denote the greatest even integer less than or equal to n , that is, n* = n or n — 1 according as n is even or odd respectively. The order of a set A , denoted by | A |, is the number of elements in A . The set whose elements are a 1 , a 2 , … , a n will be denoted by {a 1 , a 2 … , a n . The empty set will be denoted by Λ. A set will be said to include each of its elements. A set separates two elements if it includes one but not both of them. An unoriented graph U consists of two disjoint sets V ( U ), E ( U ), the elements of V ( U ) being called vertices of U and the elements of V(U) being called edges of U , together with a relationship whereby with each edge is associated an unordered pair of distinct vertices which the edge is said to join .