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

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 a1, a2 , … , an will be denoted by {a1, a2 … , an . 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.