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周二下午 1 ： 30—4 ： 15 在软件楼 4 楼密码与信息安全实验室答疑 周三下午 1 ： 15 到 3 ： 15 期中测验

周二下午 1 ： 30—4 ： 15 在软件楼 4 楼密码与信息安全实验室答疑 周三下午 1 ： 15 到 3 ： 15 期中测验. 5.2.4 Bipartite graph

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周二下午 1 ： 30—4 ： 15 在软件楼 4 楼密码与信息安全实验室答疑 周三下午 1 ： 15 到 3 ： 15 期中测验

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  1. 周二下午1：30—4：15在软件楼4楼密码与信息安全实验室答疑周二下午1：30—4：15在软件楼4楼密码与信息安全实验室答疑 • 周三下午1：15到3：15期中测验

  2. 5.2.4 Bipartite graph • Definition18: A simple graph is called bipartite if its vertex set V can be partioned into two disjoint sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex in V2. (so that no edge in G connects either two vertices in V1 or two vertices in V2).The symbol Km,n denotes a complete bipartite graph: V1 has m vertices and contains all edges joining vertices in V2, and V2 has n vertices and contains all edges joining vertices in V1. • K3,3, K2,3。 V1={x1,x2,x3,x4}, V2={y1,y2,y3,y4,y5}, or V'1={x1,x2,x3,y4,y5}, V'2={y1,y2,y3,x4},

  3. The graph is not bipartite • Theorem 5.5:A graph is bipartite iff it does not contain any odd simple circuit. • Proof:(1)Let G be bipartite , we prove it does not contain any odd simple circuit. • Let C=(v0,v1,…,vm,v0) be an simple circuit of G

  4. (2)G does not contain any odd simple circuit, we prove G is bipartite • Since a graph is bipartite iff each component of it is, we may assume that G is connected. • Pick a vertex uV,and put V1={x|l(u,x) is even simple path} ,and V2={y|l(u,y) is odd simple path} • 1)We prove V(G)=V1∪V2, V1∩V2= • Let vV1∩V2, • there is an odd simple circuit in G such that these edges of the simple circuit p1∪p2 • each edge joins a vertex of V1 to a vertex of V2

  5. 2) we prove that each edge of G joins a vertex of V1 and a vertex V2 • If it has a edge joins two vertices y1 and y2 of V2 • odd simple path • (u=u0,u1,u2,,u2n,y1,y2),even path • y2ui(0i2n) • There is ujso that y2=uj. The path (u,u1,u2,,uj-1, y2,uj+1,,u2n,y1,y2) from u to y2, • Simple path (u,u1,u2,,uj-1,y2),simple circuit (y2,uj+1,,u2n,y1,y2) • j is odd number • j is even number

  6. 5.3Euler and Hamilton paths • 5.3.1 Euler paths • Definition 19: A path in a graph G is called an Euler path if it includes every edge exactly once. An Euler circuit is an Euler path that is a circuit • Theorem 5.6: A connected multigraph has an Euler circuit if and only if each of its vertices has even degree.

  7. Proof:(1)Let connected multigraph G have an Euler circuit, then each of its vertices has even degree. • (v0,v1,…,vi, …,vk),v0=vk • First note that an Euler circuit begins with a vertex v0 and continues with an edge incident to v0, say {v0,v1}. The edge {v0,v1} contributes one to d(v0). • Thus each of G’s vertices has even degree.

  8. (2)Suppose that G is a connected multigraph and the degree of every vertex of G is even. • Let us apply induction on the number of edges of G • 1)e=1,loop The graph is an Euler circuit. The result holds 2) Suppose that result holds for em e=m+1，(G)≥2. By the theorem 5.4, there is a simple circuit C in the graph G

  9. If E(G)=E(C), the result holds • If E(G)-E(C), Let H=G-C, The degree of every vertex of H is even and e(H)m • ①If H is connected, by the inductive hypothesis, H has an Euler circuit C1， • C=(v0, v1,…,vk-1, v0) • ②When H is not connected, H has lcomponents, The degree of every vertex of components is even and the number of edges less than m. By the inductive hypothesis,each of components has an Euler circuit. Hi • G is connected

  10. the puzzle of the seven bridge in the Königsberg d(A)=3. The graph is no Euler circuit. Theorem 5.7: A connected multigraph has an Euler path but not an circuit if and only if it has exactly two vertices of odd degree. d(A)=d(D)=d(C)=3, d(D)=5 The graph is no Euler path.

  11. d(A)=d(B)=d(E)=4, d(C)=d(D)=3, • Euler path:C,B,A,C,E,A,D,B,E,D

  12. 5.3.2Hamilton paths

  13. 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.

  14. Exercise P302 1,2,3,5,6 • P306 3,4,5,6,18 • 周二下午1：30—4：15在软件楼4楼密码与信息安全实验室答疑 • 周三下午1：15到3：15期中测验 • Next: Hamiltonian paths and circuits, P304 8.3 • Shortest-path problem

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