1. Introduction to Graph Theory

2. Graph G=(V,E) V={v 1,v 2,…,v n } E={(u,v)|(u,v) is an unorder pair of V} vertex=node edge=link loop multiple edges

3. simple = no loops, no multiple edges v(G)=|V(G)| ε(G)=|E(G)| isomorphism G and H are isomorphic.

4. complete graph K n adjacent matrix v ☓ v subgraph H ⊆ G if V(H) ⊆ V(G), E(H) ⊆ E(G) spanning subgraph V(H) = V(G) degree d G (v) δ(G) △ (G) ∑ d(v)=2ε

5. path u and v are connected if there is a (u,v)-path. component G 1, G 2,… G is connected if… Corollary In any graph, the number of verticesof odd degree is even.

6. edge-transitive Cycle vertex-transitive

7. If δ ≥ 2, then G contains a cycle. If ε ≥ v, G contains a cycle. If G is simple and δ ≥ 2, then G contains a cycle of length at least δ+1. Theorem Bipartite iff no odd cycle.

8. κ(G)= minimum k for which G has a k-vertex cut G is k-connected if κ(G) ≥ k. 拿走 k 點，可能斷。但 k-1 點，一定不斷。 Connectivity κ(G)

9. κ’(G)= minimum k for which G has a k-edge cut G is k-edge-connected if κ’(G) ≥ k. 拿走 k 線，可能斷。但 k-1 線，一定不斷。 Edge-connectivity κ’(G)

10. κ=2 κ’=3 δ=4 Thm κ≤ κ’ ≤ δ

11. Hamiltonian Cycle Theorem If G is simple graph with v ≥ 3 and δ ≥ v/2, then G is hamiltonian Euler Tour Hamiltonian

12. How about bipartite graphs ? Hamiltonian laceable (Hamiltonian bi-connected) Hamiltonian connected ?

13. fault-tolerant edge-fault-tolerant pancyclic bipancyclic pan= 泛 panconnected bipanconnected fault

14. Theorem G=(X,Y,E) contains a matching that saturates every vertex in X iff |N(S)| ≥ |S| for all S ⊆ X. Coroallary If G is k-regular bipartite graph with k > 0, then G has a perfect matching. Matching Perfect matching

15. Edge chromatic number= χ’ Vizing Theorem G is simple, then χ’ = ∆ or ∆ +1 Edge coloring But …

16. Chromatic number= χ Coroallary For any graph G, χ ≤ ∆ +1 Vertex coloring

17. Theorem If G is simple, then π k (G)= π k (G-e) - π k (G ∙ e) π k (G) = chromatic polynomial e e e π k (G)= π k (G+e) + π k (G ∙ e) = +

18. = + = ++ + = k(k-1)(k-2)(k-3)+2k(k-1)(k-2)+k(k-1) = k(k-1)(k 2 -3k+3) If k=0 … If k=1… If k=2… If k=3…

19. 多謝

20. ∈ ∋ ∉ ∌ ⇐ ⇒ ⇔ ⇍ ⇎ ⇏ ≤ ≥ ≠ ⊕ ⊗ ⊖  ⌀ → ← ↝ ↜ ↛ ↚ ∘ ∙ ☓ ⊆ ⊇ ⊄ ⊅ ⊈ ⊉ ⊃ ⊂ ∪ ∩ ∨ ∧ ∃ ∄ ∞ ∀ ∆ ∇ π ∏ ∑ ∓ ⊠ ⌯ ↺ ↻