1. 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

2. 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

3. 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

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

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

14. 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

15. 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

16. 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, and
Theorem 4.2.2
determine
Discrete Mathematics
Yuan Luo Computer Science and Engineering Department Shanghai

17. 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

18. 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

19. 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

20. 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

21. 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

24. 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 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

25. 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 at page 80 of Chinese Textbook.
Hint:
𝑑 ∗ = 𝑚 ∗ − 𝑛 ∗ +𝟐=𝑚−𝑑+𝟐=𝑚−(𝑚−𝑛+k+𝟏)+𝟐=n−k+1

26. 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

27. 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.

28. 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.

29. 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