<P> For instance, as C.A.B. Smith proved, in any cubic graph G there must be an even number of Hamiltonian cycles through any fixed edge uv; Thomason (1978) used a proof based on the handshaking lemma to extend this result to graphs G in which all vertices have odd degree . Thomason defines an exchange graph H, the vertices of which are in one - to - one correspondence with the Hamiltonian paths beginning at u and continuing through v. Two such paths p and p are connected by an edge in H if one may obtain p by adding a new edge to the end of p and removing another edge from the middle of p; this is a symmetric relation, so H is an undirected graph . If path p ends at vertex w, then the vertex corresponding to p in H has degree equal to the number of ways that p may be extended by an edge that does not connect back to u; that is, the degree of this vertex in H is either deg (w) − 1 (an even number) if p does not form part of a Hamiltonian cycle through uv, or deg (w) − 2 (an odd number) if p is part of a Hamiltonian cycle through uv . Since H has an even number of odd vertices, G must have an even number of Hamiltonian cycles through uv . </P> <P> In connection with the exchange graph method for proving the existence of combinatorial structures, it is of interest to ask how efficiently these structures may be found . For instance, suppose one is given as input a Hamiltonian cycle in a cubic graph; it follows from Smith's theorem that there exists a second cycle . How quickly can this second cycle be found? Papadimitriou (1994) investigated the computational complexity of questions such as this, or more generally of finding a second odd - degree vertex when one is given a single odd vertex in a large implicitly - defined graph . He defined the complexity class PPA to encapsulate problems such as this one; a closely related class defined on directed graphs, PPAD, has attracted significant attention in algorithmic game theory because computing a Nash equilibrium is computationally equivalent to the hardest problems in this class . </P> <P> The handshaking lemma is also used in proofs of Sperner's lemma and of the piecewise linear case of the mountain climbing problem . </P>

Show that the number of odd degree vertices in (any) graph g is even