Discrete Mathematics Graph: Planar Graph Yuan Luo Introduction Graph Coloring Discrete Mathematics Graph: Planar Graph Yuan Luo Computer Science and Engineering Department Shanghai Jiao Tong University Fall 2015 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Euler’s Formula (点边面关系) Introduction Graph Coloring Content 1 Introduction Euler’s Formula (点边面关系) Properties Analogy (面边关系)to the Handshaking (点边关系) Maximal Planar Graphs (simple graph, why ?? ) Simple Planner Graphs with 𝑑 ≥2 Nonplanar Graphs Dual Graphs Graph Coloring 2 Remark. 没有新的点面关系 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Content 1 Introduction Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Content 1 Introduction Euler’s Formula Properties Analogy to the Handshaking Maximal Planar Graphs Simple Planner Graphs with 𝑑 ≥2 Nonplanar Graphs Dual Graphs Graph Coloring 2 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Definition Planar Graph A graph is called planar if it can be drawn in the plane without any edges crossing.

Definition Region A planar representation of a graph splits the plane into regions, including an unbounded region. The number of regions is denoted by d (d=6 in this example). Two regions only having common vertices are not considered to be adjacent.

Euler Formula A. Three versions of Euler’s Formula: 点面边的计数关系 Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Euler Formula A. Three versions of Euler’s Formula: 点面边的计数关系 Connected planar graph : 𝒅=𝒎−𝒏+𝟐 Planar graph with k connected components: 𝒅=𝒎−𝒏+𝒌+𝟏 Planar graph : 𝑑 ≥𝑚−𝑛+2 Note that the graph is planar if and only if each component is planar. So connected graph is important in the research of planar graph ! Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Proof. Connected planar graph 𝑑=𝑚−𝑛+2 subgraph Number of edges Number of regions Spanning Tree n-1 1 (add edge one by one) Original Graph m 𝑑=𝑚−𝑛+2

𝒅= 𝒅𝟏−𝟏 +…+ 𝒅𝒌−𝟏 +𝟏 𝒎=𝒎𝟏+…+ 𝒎 𝒌 𝒏=𝒏𝟏+…+ 𝒏 𝒌 Planar graph with k connected components 𝒅=𝒎−𝒏+𝒌+𝟏 Proof. Consider each component respectively. 𝒅𝟏=𝒎𝟏−𝒏𝟏+2 ⋮ 𝒅𝒌= 𝒎 𝒌 − 𝒏 𝒌 +𝟐 𝒅= 𝒅𝟏−𝟏 +…+ 𝒅𝒌−𝟏 +𝟏 𝒎=𝒎𝟏+…+ 𝒎 𝒌 𝒏=𝒏𝟏+…+ 𝒏 𝒌

Properties Analogy to the Handshaking Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Content 1 Introduction Euler’s Formula Properties Analogy to the Handshaking Maximal Planar Graphs Simple Planner Graphs with 𝑑 ≥2 Nonplanar Graphs Dual Graphs Graph Coloring 2 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Properties Analogy to Handshaking Theorem Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Properties Analogy to Handshaking Theorem planar graph with no cut edges: the number of all edges on region boundaries is 2𝑚. (B. 面边握手) planar graph with cut edges: the number of all edges on region boundaries is < 2m. The number of edges around a region r is also called the degree of the region r ! 好像自环对点度贡献2 If the contribution of a cut edge (to the degree of the region) is counted twice, < should be = Better than Chinese textbook in dealing with cut edge. Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

利用面边握手 利用点边握手 Remark. Totally, we have 𝑑=𝑚−𝑛+𝑘+1 2𝑚= 𝑣 deg⁡(𝑣) , 𝑣 𝑚𝑒𝑎𝑛𝑠 𝑣𝑒𝑟𝑡𝑒𝑥 2𝑚= 𝑟 deg⁡(𝑟) , 𝑟 𝑚𝑒𝑎𝑛𝑠 𝑟𝑒𝑔𝑖𝑜𝑛

Content Euler’s Formula Maximal Planar Graphs Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Content 1 Introduction Euler’s Formula Properties Analogy to the Handshaking Maximal Planar Graphs Simple Planner Graphs with 𝑑 ≥2 Nonplanar Graphs Dual Graphs Graph Coloring 2 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Maximal Planar Graph (define in simple graph) Consider a planar graph G. If there are no nonadjacent vertices u and v such that 𝑮+(𝒖,𝒗) can be drawn in planar, then G is called maximal. If there exist nonadjacent vertices u and v such that 𝑮+(𝒖,𝒗) can be drawn in planar, then G is not maximal.

Any two of them decide all others. Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Maximal Planar Graphs (simple graph) C. Properties (for n≥3): It is connected, has no cut edge, and the number of edges on the boundary of each region is 3 (See Dai’s book pp.71 Graph 4.4: 若≥4，𝑖1,𝑖3和𝑖2,𝑖4必有一对点不邻，矛盾。) Important formulas: (See Dai’s book Theorem 4.2.1) : (1) 𝑚=3𝑛 −6. (极大) (2) 𝑑=2𝑛 −4. (极大) (3) 𝟑𝒅=𝟐𝒎. (极大；面边握手) (4) 𝒅=𝒎 −𝒏+𝟐. (平面连通图；欧拉) Any two of them decide all others. Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Content Euler’s Formula Simple Planner Graphs with 𝑑 ≥2 Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Content 1 Introduction Euler’s Formula Properties Analogy to the Handshaking Maximal Planar Graphs Simple Planner Graphs with 𝑑 ≥2 Nonplanar Graphs Dual Graphs Graph Coloring 2 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Simple Planner Graphs with 𝑑≥2 Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Simple Planner Graphs with 𝑑≥2 D. Important formulas: 𝑚≤3𝑛 −6. 𝑑≤2𝑛 −4. 𝟑𝒅≤𝟐𝒎 (面边握手). 𝒅≥𝒎 −𝒏+𝟐 (欧拉). Also see Dai’s book Example 4.2.1, 4.2.2, 4.2.3, 4.2.4 and Theorem 4.2.2 determine Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Content Euler’s Formula Nonplanar Graphs Dual Graphs 1 Introduction Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Content 1 Introduction Euler’s Formula Properties Analogy to the Handshaking Maximal Planar Graphs Simple Planner Graphs with 𝑑 ≥2 Nonplanar Graphs Dual Graphs Graph Coloring 2 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Graph G is planar if and only if G has no K-types subgraphs. Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Nonplanar Graphs 𝐾 5 (𝐾 (1) ), 𝐾 3,3 𝐾 2 , 𝐾−𝑡𝑦𝑝𝑒𝑠 ⋯ are not planar 𝐾 5 : 𝒎=𝟏𝟎, 𝒏=𝟓, 𝒎≤𝟑𝒏−𝟔 𝒊𝒔 𝒏𝒐𝒕 𝒔𝒂𝒕𝒊𝒔𝒇𝒊𝒆𝒅; 𝐾 3,3 : 𝒎=𝟗, 𝒏=𝟔, 𝒅=𝟓, 𝒎≤𝟑𝒏−𝟔 𝒊𝒔 𝒔𝒂𝒕𝒊𝒔𝒇𝒊𝒆𝒅 𝒃𝒖𝒕 ′ 𝒃 𝒊𝒑𝒂𝒓𝒕𝒊𝒕 𝒆 ′ 𝒎𝒆𝒂𝒏𝒔 𝒏𝒐 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆, 𝒊.𝒆. 𝒅𝒆𝒈 𝒓 ≥𝟒, 𝒄𝒐𝒏𝒕𝒓𝒂𝒅𝒊𝒄𝒕𝒊𝒐𝒏; 𝐾−𝑡𝑦𝑝𝑒𝑠: with more vertices on the edge of 𝐾 5 , 𝐾 3,3 Kuratowski Theorem: Graph G is planar if and only if G has no K-types subgraphs. Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Nonplanar graphs Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Nonplanar graphs Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Content Euler’s Formula Dual Graphs 1 Introduction Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Content 1 Introduction Euler’s Formula Properties Analogy to the Handshaking Maximal Planar Graphs Simple Planner Graphs with 𝑑 ≥2 Nonplanar Graphs Dual Graphs Graph Coloring 2 Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Content The definition and drawing program of dual graph. (English Textbook the 6th Edition: p.667) Each region of the map is represented by a vertex. Edges connect two vertices if the regions represented by these vertices have a common border. Two regions that touch at only one point are not considered adjacent. The resulting graph is called the dual graph of the map. Background in graph coloring. The problem of coloring the regions of a map is equivalent to the problem of coloring the vertices of the dual graph so that no two adjacent vertices in this graph have the same color. Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Dual Graphs (See Dai’s book pp.79 Property 4.5.1 and Theorem 4.5.1) Euler’s Formula Properties Analogy to the Handshaking Theorem Introduction Maximal Planar Graphs Graph Coloring Simple Planner Graphs with d ≥2 Nonplanar Graphs Dual Graphs Dual Graphs Graph G has a dual graph if and only if G is planar. (See Dai’s book pp.79 Property 4.5.1 and Theorem 4.5.1) If G is a planar graph, 𝒕𝒉𝒆𝒏 𝑮 ∗ is connected and 𝑮 ∗ = 𝑮 ∗∗∗ = 𝑮 ∗∗∗∗∗ =⋯, 𝑮 ∗∗ = 𝑮 ∗∗∗∗ = 𝑮 ∗∗∗∗∗∗ =⋯ . If G is a connected planar graph, then 𝑮= 𝑮 ∗∗ Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Assume that G is a planar graph with k connected components, the relations between 𝐺 and 𝐺 ∗ are: (1) 𝑚 ∗ =𝑚. (2) 𝑛 ∗ =𝑑. (3) 𝑑 ∗ =n−k+1. This is a generalization of the Property 4.5.4 at page 80 of Chinese Textbook. Hint: 𝑑 ∗ = 𝑚 ∗ − 𝑛 ∗ +𝟐=𝑚−𝑑+𝟐=𝑚−(𝑚−𝑛+k+𝟏)+𝟐=n−k+1

Introduction Graph Coloring Graph Coloring The Four Color Theorem: The chromatic number of a planar graph is no greater than four. ( See textbook pp.668 Theorem 1) Discrete Mathematics Yuan Luo Computer Science and Engineering Department Shanghai

Chinese Textbook Ch. 4 Exercises 2. In maximal planar graph with 𝒏≥𝟒, for each vertex 𝒗 , 𝐝𝐞𝐠𝐫𝐞𝐞(𝒗)≥𝟑. Proof. First, assume that there exists 𝑣 0 such that degree 𝑣 0 =1, which is impossible since there is no cut-edge in maximal planar graph. Second, assume that there exists 𝑣 0 such that degree( 𝑣 0 )=2. Deleting 𝑣 0 , since 𝑛−1≥3, we have 𝑚−2≤3 𝑛−1 −6, 𝑖.𝑒. 𝑚≤3𝑛−7, which is impossible since 𝑚=3𝑛−6.

Chinese Textbook Ch. 4 Exercises 8. In simple planar graph with 𝑛≥4, there exist at least 4 vertices with 𝑑𝑒𝑔𝑟𝑒𝑒≤ 5. Proof. Assume that there exist at most 3 vertices with degree ≤ 5. Let 𝛿 denote the minimum degree of the vertices. 𝜹≥𝟑: 𝐇𝐚𝐧𝐤𝐬𝐡𝐚𝐤𝐢𝐧𝐠 2𝑚≥3×3+ 𝑛−3 ×6=6𝑛−9 , 𝑖.𝑒. 𝑚≥3𝑛−4.5 which is impossible since 𝑚≤3𝑛−6 (Planar graph). 𝜹=𝟐: 𝐇𝐚𝐧𝐤𝐬𝐡𝐚𝐤𝐢𝐧𝐠 2𝑚≥3×2+ 𝑛−3 ×6=6𝑛−12. 𝐏𝐥𝐚𝐧𝐚𝐫 𝐠𝐫𝐚𝐩𝐡 𝑚≤3𝑛−6. So this is a maximal planar graph with 3 points of degree 2, which is in contradiction to Exercise 2.

𝐇𝐚𝐧𝐤𝐬𝐡𝐚𝐤𝐢𝐧𝐠 2𝑚≥ 𝑛−3 ×6, 𝑖.𝑒. 𝑚≥3𝑛−9. 𝜹=𝟏:Eliminate a point with degree 1. 𝐇𝐚𝐧𝐤𝐬𝐡𝐚𝐤𝐢𝐧𝐠 2(𝑚−1)≥2×1 + 𝑛−3 ×6−𝟏, 𝑖.𝑒. 𝑚−1≥3 𝑛−1 −5.5 which is impossible since 𝑚−1≤3(𝑛−1)−6 (Planar graph). 𝜹=𝟎: Eliminate a point with degree 0. 𝐇𝐚𝐧𝐤𝐬𝐡𝐚𝐤𝐢𝐧𝐠 2𝑚≥ 𝑛−3 ×6, 𝑖.𝑒. 𝑚≥3𝑛−9. 𝐏𝐥𝐚𝐧𝐚𝐫 𝐠𝐫𝐚𝐩𝐡 𝑚≤3 𝑛−1 −6=3𝑛−9. “2 points with degree 0” “maximal planar graph” 矛盾