Graph Theory Ming-Jer Tsai

Outline Graph Graph Theory Grades Q & A

Graph A triple consisting of a vertex set V(G), an edge set E(G), and a relation that associates with each edge two vertices (not necessary distinct) called its endpoints. e1e1 e2e2 e6e6 e5e5 e3e3 e7e7 e4e4 x yw z

Graph Theory Matching Connectivity Coloring Planar Graphs Hamiltonian Cycles

Matching Matching: A matching in a graph G is a set of non-loop edges with no shared endpoints

(Hall’s Condition) An X,Y-bigraph G has a matching that saturates X if and only if |N(S)|>=|S| for all S  X. N(S): the set of vertices having a neighbor in S. Matching BCDEA X Y S = {B, D, E}

Algorithm for Maximum Bipartite Matching Algorithm for Maximum Weighted Bipartite Matching Matching Edge weight W i,j u1u1 u2u2 u3u3 v1v1 v2v2 v3v3 u1u1 u2u2 u3u3 v3v3 v2v2 v1v1 w(M)=6+5+8=19

(Tutte’s Condition) A graph G has a perfect matching if and only if o(G-S)<=|S| for every S  V(G). o(G-S): the number of components of odd orders in G-S. Matching S Odd component Even component

Connectivity For a simple graph G,  (G)<=  ’(G)<=  (G).  (G): the minimum size of a vertex set S such that G-S is disconnected or has only one vertex (connectivity of G).  ’(G): the minimum size of a set of edges F such that G-F has more than one component (edge-connectivity of G).  (G): minimum degree of G. 1.  (G) =  ’(G) =  (G) = 3.

( Menger Theorem ) If x,y are vertices of a graph G and xy  E(G),  (x,y) = (x,y).  (x,y): the minimum size of a set S  V(G)-{x,y} such that G-S has no x,y-path. (x,y): the maximum size of a set of pairwise internally disjoint x,y-paths. Connectivity

(Brook’s Theorem) If G is a connected graph other than a complete graph or an odd cycle,  (G)<=  (G).  (G): The least k such that G is k-colorable.  (G): the maximum degree in G. Coloring

(Whitney’s Theorem) The number of proper k- coloring of G is  S  E(G) (-1) |S| k c(G(S)). c(G): the number of components of a graph G. G(S): the spanning subgraph of G with edge set S  E(G). The number of proper k-coloring of a kite is k 4 -5k 3 +10k 2 -(2k 2 +8k 1 )+5k-k.

Edge-Coloring (Vizing and Gupta’s Theorem) If G is a simple graph, x’(G) ≤ Δ(G)+1.  ’(G): The least k such that G is k-edge-colorable.

Planar Graph (Kuratowski’s Theorem) A graph is planar iff it does not contain a subdivision of K 5 or K 3,3.

(Chvatal’s Condition) Let G be a simple graph with vertex degree d 1 ≤ … ≤ d n, where n ≥ 3. If i i or d n-i ≥ n-i, G has a hamiltonian cycle. Hamiltonian Cycles

(Four Color Theorem) Every planar graph is 4- colorable. Four Color Theorem

Grades 3 Exams (60%) Report (20%) Presentation and Discussion (10%) Attendance and Discussion (10%)

Q & A