5 . 3 . 2 Hamilton paths

5.3.2Hamilton paths

Definition 20: A Hamilton paths is a path that contains each vertex exactly once. A Hamilton circuit is a circuit that contains each vertex exactly once except for the first vertex, which is also the last.

Theorem 5.8: Suppose G(V,E) that has a Hamilton circuit, then for each nonempty proper subset S of V(G), the result which (G-S)≤|S| holds, where G-S is the subgraph of G by omitting all vertices of S from V(G). (G-S)=1，|S|=2 The graph G has not any Hamilton circuit, if there is a nonempty purely subgraph S of G so that (G-S)>|S|.

Omit {b,h,i} from V, • (G-S)=4>3=|S|，The graph has not any Hamilton circuit

If (G-S)≤|S| for each nonempty proper subset S of G, then G has a Hamilton circuit or has not any Hamilton circuit. • For example: Petersen graph

Proof: Let C be a Hamilton circuit of G(V,E). Then (C-S)≤|S| for each nonempty proper subset S of V • Why? • Let us apply induction on the number of elements of S. • |S|=1, • The result holds • Suppose that result holds for |S|=k. • Let |S|=k+1 • Let S=S'∪{v}，then |S'|=k • By the inductive hypothesis, (C-S')≤|S'| • V(C-S)=V(G-S) • Thus C-S is a spanning subgraph of G-S • Therefore (G-S)≤(C-S)≤|S|

Theorem 5.9: Let G be a simple graph with n vertices, where n>2. G has a Hamilton circuit if for any two vertices u and v of G that are not adjacent, d(u)+d(v)≥n. n=8,d(u)=d(v)=3, u and v are not adjacent,d(u)+d(v)=6<8, But there is a Hamilton circuit in the graph. Note:1)if G has a Hamilton circuit , then G has a Hamilton path Hamilton circuit :v1,v2,v3,…vn,v1 Hamilton path:v1,v2,v3,…vn, 2)If G has a Hamilton path, then G has a Hamilton circuit or has not any Hamilton circuit

Corollary 1: Let G be a simple graph with n vertices, n>2. G has a Hamilton circuit if each vertex has degree greater than or equal to n/2. • Proof: If any two vertices of G are adjacent ,then G has a Hamilton circuit v1,v2,v3,…vn,v1。 • If G has two vertices u and v that are not adjacent, then d(u)+d(v)≥n. • By the theorem 5.9, G has a Hamilton circuit. • Kn has a Hamilton circuit where n≥3

Theorem 5.10: Let the number of edges of G be m. Then G has a Hamilton circuit if m≥(n2-3n+6)/2,where n is the number of vertices of G. • Proof: If any two vertices of G are adjacent ,then G has a Hamilton circuit v1,v2,v3,…vn,v1. • Suppose that u and v are any two vertices of G that are not adjacent. • Let H be the graph produced by eliminating u and v from G. • Thus H has n-2 vertices and m-d(u)-d(v) edges.

Theorem 5. 11：Let G be a simple graph with n vertices, n>2. G has a Hamilton path if for any two vertices u and v of G that are not adjacent, d(u)+d(v)n-1.

5.4 Shortest-path problem • Let G=(V,E,w) be a weighted connected simple graph, w is a function from edges set E to position real numbers set. We denoted the weighted of edge {i,j} by w(i,j), and w(i,j)=+ when {i,j}E • Definition 21: Let the length of a path p in a weighted graph G =(V,E,w) be the sum of the weights of the edges of this path. We denoted by w(p). The distance between two vertices u and v is the length of a shortest path between u and v, we denoted by d(u,v).

Dijkstra’s algorithm(E.W.Dijkstra) • In 1959

Let G=(V,E,w) and |V|=n where w>0. Suppose that S is a nonempty subset of V and v1S. Let T=V-S. Example: Suppose that (u,v',v'',v''',v) is a shortest path between u and v. Then (u,v',v'',v''') is a shortest path between u and v'''.

Exercise P306 3,4,5,6,18 • Next: Shortest-path problem • Trees and their properties 7.4 P273

5 . 3 . 2 Hamilton paths