Comprehensive Overview of Graph Theory Concepts

1. Introduction to Graph Theory

2. Graph G=(V,E) V={v1,v2,…,vn} vertex=node E={(u,v)|(u,v) is an unorder pair of V} 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 Kn adjacent matrix v☓v subgraph H ⊆ G if V(H) ⊆ V(G),E(H) ⊆ E(G) spanning subgraph V(H) = V(G) degree dG(v) δ(G) △(G) ∑d(v)=2ε

5. Corollary In any graph, the number of verticesof odd degree is even. path u and v are connected if there is a (u,v)-path. component G1, G2,… G is connected if…

6. vertex-transitive edge-transitive Cycle

7. Theorem Bipartite iff no odd cycle. • 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.

8. Connectivity κ(G) 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。

9. Edge-connectivity κ’(G) 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。

10. Thm κ≤ κ’ ≤ δ κ=2 κ’=3 δ=4

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

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

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

14. Matching Perfect matching 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.

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

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

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

18. = + + = + + = k(k-1)(k-2)(k-3)+2k(k-1)(k-2)+k(k-1) = k(k-1)(k2-3k+3) If k=0 …If k=1…If k=2…If k=3…

19. 多謝

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