on the planarity of graphs 2

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Chapter 1

INTRODUCTION

Situation Analysis

Mathematics is used throughout the world as an essential tool in

many fields. Its many applications towards improving human life became

a very important role. It arises from many different kinds of problems

and many problems also arise within it. But the many properties and

concepts that Mathematics contains make it difficult to understand.

Mathematics can, broadly speaking, be subdivided into the study of

quantity, structure, space, and change.

An article on “Graph Theory” from wikepedia.com (2010) pinions

graph theory as the study of mathematical structures utilized to project

pairwise relations between objects from a collection. In here, the term

“graph” refers to a collection of vertices or nodes, and a collection of

edges connecting pairs of vertices.

The same article further expounded that graphs are among the

most ubiquitous models of both natural and human-made structures.

They can be used to model many types of relations and process dynamics

in physical, biological and social systems.

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Many problems of practical interest can be represented by graphs.

In computer science, they are used to represent networks of

communication, data organization, computational devices, the flow of

computation, and the likes. In Mathematics, they are useful in Geometry

and certain parts of Topology (“Graph Theory,” 2010).

In the problems on vehicle routing, telecommunications, laying out

circuits on computer chips and even in solving a puzzle game, a special

kind of graph called planar graph is used.

Planar graphs are guided by numerous theorems that are

somewhat complicated and difficult to derive and prove. Like the

Kuratowski, Wagner, Euler theorems and others, lots of methods are

used in determining a graph if it is planar or not. But they are too

confusing that they seem to hinder one’s ability to understand and

master the concepts.

The concepts and theorems about the planarity of graphs are,

indeed, quite hard to understand. Thus, in this thesis, concepts and

theorems will be explained and proven in a simplified manner in an effort

to shed light on the foregoing concepts to the readers.

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This thesis will not just explain what planar graphs are but will

also tell the readers the necessary details to be learned in order to

master the concepts and theorems of these graphs and try to relate its

applications and relevance in the lives of the readers.

Objectives

This thesis about the planarity of graphs will be conducted to:

1.Formulate other methods or ways to verify the planarity of

graphs.

2.Present and exemplify some special properties of planar

graphs.

3.Provide simplified proofs to theorems and formulas on

planar graphs.

4.Formulate and illustrate new propositions related to planar

graphs.

5.Cite probable applications or uses of Planarity of Graphs in

everyday life.

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Significance of the Study

Mathematics is really a broad and complicated field of knowledge

concerning numbers and figures which causes lots of confusions to

students as well as to teachers. Nevertheless, these complications serve

as an impetus for the conduct of researches for purposes of exploration

and advancing the frontiers of knowledge.

This thesis may be beneficial to the following:

Students. This thesis may help the students in solving problems

related to graphs easier and faster. It may also promote a greater

understanding to students regarding the different properties and

principles that lie within graphs.

Teachers. Teachers may use this study as a reference in shaping

the minds of students about graphs. They may also use the formulas and

proofs that will be discussed in the later part as an easier way of

teaching students on how to prove the theorems about the planarity of

graphs.

Future Researchers.Future researchers may use this thesis as a

reference in making other researches related to graphs. This will guide

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them to the prevalence of the different information regarding graphs

especially on the planarity of graphs.

School Administrators. This thesis may help in raising the

school’s fame due to possible improvement of students in their

mathematical skills thus can make them capable of winning

competitions.

Scope and Delimitation

This research mainly focuses on the planarity of graphs because

they involve more complicated properties, theorems, and formulas.

This will involve the basic ideas about graphs for it is required to

be mastered first before tackling more integrated ideas to avoid

difficulties. It will also provide a suitable background and a clear view on

planar graphs in order for the readers to be familiarized on the properties

and the reasons why they were considered as planar. The researchers will

present simplified proofs to theorems and properties of planar graphs,

and formulate new propositions in relation to the study.

Different theorems and formulas on the planarity of graphs will be

discussed thoroughly along with their respective proofs and derivations.

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Related problems with answers and complete solutions will also be

included that will bring the readers the important ideas necessary in

order to master the planarity of graphs even the most confusing ones in a

simplified way.

Definition of Terms

Bipartite Graph.It is a graph whose vertex set can be partitioned

into two subsets X and Y, so that each edge has one end in X and one

end in Y.

Boundary.It is a subgraph that is formed by the vertices and

edges that are incident with a region.

Complete Graph.It is a simple graph in which each pair of

distinct vertices is joined by an edge.

Complete bipartite graph.It is a simple bipartite graph with

bipartition (X, Y) in which each vertex in X is joined to each vertex in Y.

Cycle. The edge set of an undirected closed path without repeated

vertices or edges. This may also be called a circuit, circle, or polygon

(“Cycle (Graph Theory)”, 2010).

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Degree. The degree of a vertex of a graph is the number of edges

incident to the vertex, with loops counted twice. The degree of a vertex v

is denoted deg (v).

Edge.It is a line or curve that connects an unordered pair of

vertices.

Embeddings. An embedding of a graph into a surface is a drawing

of a graph on the surface in such a way that its edges may intersect only

at their endpoints (“Graph Embedding,” 2010).

Graph.It is an abstract representation of a set of objects where

some pairs of the objects are connected by links. The interconnected

objects are represented by mathematical abstractions called vertices, and

the links that connect some pairs of vertices are called edges (“Graph

(Mathematics),” 2010).

Homeomorphism.In graph theory, two graphs G and G' are

homeomorphic if there is an isomorphism from some subdivision of G to

some subdivision of G'.

Inductive proof.Formal method of proof in which the proposition

P(n + 1) is proved true on the hypothesis that the proposition P(n) is true

(“Inductive Proof,” 2009).

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Isomorphic Graphs. These are graphs that contain the same

number of graph vertices connected in the same way (Weisstein, 2010).

Maximal planar graph.It is a planar graph to which no new arcs

can be added without forcing crossings and hence violating planarity.

Minor. A graph H is a minor of a graph G if a copy of H can be

obtained from G by deleting and/or contracting edges of G.

Planar graph.It is a graph that can be drawn in a plane without

any of its edges intersecting.

Plane.In mathematics, a plane is any flat, two-dimensional

surface (“Plane (Geometry).” 2010).

Plane graph.It is a planar graph that is drawn in the plane so

that no two edges intersect.

Proof.It is an organized process which shows the validity of a

certain theorem by giving mathematical statements and reasons for every

statement in a theorem.

Region.It is the connected pieces of the plane that remain when

the vertices and edges of a graph are removed. It is sometimes called

face.

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Subdivision. A subdivision of a graph G is a graph resulting from

the subdivision of edges in G.

Subgraph.It is a graph whose graph vertices and graph edges

form subsets of the graph vertices and graph edges of another graph

(Weisstein, 2010).

Supergraph.If a graph A is a subgraph of graph B, then graph B

is said to be a supergraph of graph B.

Theorem.In mathematics, a theorem is a statement that has been

proven on the basis of previously established statements (“Theorem,”

2010).

Vertex.In graph theory, a vertex (plural vertices) or node is the

fundamental unit out of which graphs are formed (“Vertex (Graph

Theory),” 2010).

Mathematical Symbols and Notations

v – denotes the number of vertices

e – denotes the number of edges

r – denotes the number of regions

V(G) – the set of vertices of a graph G

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E(G) – the set of edges of a graph G

ΨG - incidence function of a graph G

Kn – complete graph with n vertices

Km,n – complete bipartite graph with bipartition (m,n) in which

each vertex in m is joined to each vertex in n.

Σ deg Vi – summation of all the degrees

≤ - “less than or equal to”

≥ - “greater than or equal to”

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Chapter 2

REVIEW OF LITERATURE

Different mathematicians all over the world have contributed their

knowledge on graphs to improve everybody’s point of view about the

complicated world of mathematics. One of their contributions is the

discovery of graphs that is used to model both natural and human-made

structures. And due to several arising problems on the subject, some of

these mathematicians discovered a special graph known as planar graph

which serves as a solution to various practical problems. Planar graphs

have complicated properties and theorems that are hard to understand

that is why many researchers conduct related studies and never give up

sharing their ideas on the planarity of graphs.

The planarity of graphs was adopted from the theorems made by

well known mathematicians such as Euler. The different properties of

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these graphs were also studied and proven by some mathematicians in

order to be valid and reliable.

Some authors of mathematics books also gave their support in

order to provide more reliable sources of information regarding the

planarity of graphs. Some of these books were all about Discrete

Mathematics and Graph Theory, and they contain the details and facts

about the planarity of graphs. Most of these details, facts and important

information were combined, enhanced and published in order to provide

more systematic references for the planarity of graphs. Examples of these

books were Discrete Mathematics and Its Applications, Sixth Edition, by

K. H. Rosen, published in 2007, Introduction to Graph Theory, by R. J.

Trudeau, published in 1993, The Theory of Graphs and its Applications

by C. Berge, published in 1958 and Introductory Graph Theory, by G.

Chartrand, 1977. The authors of these books made an outstanding effort

in presenting different proofs to some theorems on the planarity of

graphs. They considered the three houses and three utilities problem

wherein each house will be connected to each of the utilities which by

tradition are gas, water, and electricity. Is it possible to join these houses

and utilities so that none of the connections cross? This problem can be

modeled using the complete bipartite graph K3,3.

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Douglas B. West in the second edition of his book entitled

“Introduction to Graph Theory” published in 2001 discussed on the

characterization of planar graphs which is the Kuratowski’s Theorem. He

mentioned that the graphs K5 and K3,3 are the crucial graphs that lead to

the characterization of planar graphs known as Kuratowski’s Theorem.

V. K. Balakrishnan in his book “Shaum’s Outlines on Graph

Theory” discussed on graph embeddings with the theorems and provided

about a hundred of problems on planarity on graphs with their

respective answers with solutions. He discussed that such drawings of a

planar graph in a plane is a planar embedding of the graph. A plane

graph is a particular representation of a planar graph in the plane drawn

in such a way that any pair of edges meets only at their end vertices.

Some books and studies were also published by several authors.

One of these which is present in the internet is the study of D. Dolev, T.

Leighton, and H. Trickey entitled “Planar Embedding of Planar Graphs.”

IMPORTANT CONCEPTS

A.Preliminary Concepts

I.Graphs

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A graph refers to a collection of vertices or 'nodes' and a

collection of edges that connect pairs of vertices. A graph may be

undirected, meaning that there is no distinction between the two

vertices associated with each edge, or its edges may be directed from

one vertex to another (“Graph (Mathematics),” 2010).

A graph G is an ordered triple denoted by V(G), E(G), ΨG

consisting of a non empty set V(G) of vertices, a set E(G) of edges,

and an incidence function ΨG that associates with each edge of G to

an unordered pair of vertices of G. There is no unique way of

drawing a graph, the relative positions of points representing the

vertices and lines representing the edges have no significance, and

thus we can draw a graph in infinite number of ways.

Special Classes of Graphs

1. Complete graph (kn) – it is a simple graph in which each

pair of distinct vertices is joined by an edge.

2. Empty/ null graph – it is a graph with no edges.

3. Bipartite graph – it is a graph whose vertex set can be

partitioned into two subsets X and Y, so

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that each edge has one end in X and one

end in Y, such partition (X,Y) is called a

bipartition of the graph.

4. Complete bipartite graph (km,n) – it is a simple bipartite

graph with bipartition (X,Y) in which each

vertex of X is joined to each vertex of Y.

II. Isomorphic Graphs

Two graphs G and H are identical (G H) if V(G) = V(H), E(G) =≡

E(H) and ΨG = ΨH. They can be represented by identical diagram

A B 1 2

F C 6 3

E D 5 4

G H

However, two graphs that are not identical may have the same

diagram. Such graphs are called isomorphic graphs. Two graphs G

and H are said to be isomorphic if there are bijections: θ: V(G) V(H)

and Φ: E(G)E(H) such that ΨG(e) = uv if and only if ΨHΦ(e) =

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θ(u)θ(v); such that a pair (θ,Φ) of mappings is called an isomorphism

between G and H.

To show that two graphs are isomorphic, one must indicate an

isomorphism between them. The pair of mappings (θ,Φ) defined by

θ(v1) = y θ(v2) = x θ(v3) = u θ(v4) = v θ(v5) = w

and

Φ(e1) = h Φ(e2) = g Φ(e3) = b Φ(e4) = a

Φ(e5) = e Φ(e6) = c Φ(e7) = d Φ(e8) = f

is an isomorphism between the graphs G and H. graphs G and

H have the same structure and differ only in the names of vertices

and edges. 1

A B

F C 5 6

4

E D 3 2

G H

θ(A) = 1 θ(B) = 2 θ(C) = 3 θD) = 4 θ(E) = 5 θ(F) = 6

and

Φ(AB) = 12 Φ(AC) = 13 Φ(AE) = 15 Φ(AF) = 16

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Φ(BF) = 26 Φ(BD) = 24 Φ(BC) = 23 Φ(CD) = 34

Φ(CE) = 35 Φ(DE) = 45 Φ(DF) = 46 Φ(EF) = 56

These concepts on graphs and isomorphic graphs including the

illustrations are extracted from the discussions of Dr. Raquel D.

Quiambao, the chairman of the Mathematics Department in the

College of Sciences at DMMMSU-SLUC, Agoo, La Union, in her

Graph Theory class in the school year 2009 – 2010 at DMMMSU-

SLUC.

B. Primary Concepts

I. Planar Graphs

In graph theory, a planar graph is a graph that can be

embedded in a plane so that no edges intersect. The Polish

mathematician Kazimierz Kuratowski provided a characterization of

planar graphs, now known as Kuratowski's theorem:

A finite graph is planar if and only if it does not contain a

subgraph that is an expansion of K5 (the complete graph on 5

vertices) or K3,3 (complete bipartite graph on six vertices, three of

which connect to each of the other three); A finite graph is planar if

and only if it does not contain a subgraph that is homeomorphic to

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K5 or K3,3 and; A finite graph is planar if and only if it does not have

K5 or K3,3 as a minor.

K5 K3,3

In practice, it is difficult to use Kuratowski's criterion to quickly

decide whether a given graph is planar. However, there exist fast

algorithms for this problem: for a graph with v vertices, it is possible

to determine in time O(v) whether the graph is planar or not. For a

simple, connected, planar graph with v vertices ande edges:

Theorem 1. If v ≥ 3 then e ≤ 3v - 6

Theorem 2. If v > 3 and there are no cycles of length 3, then

e ≤ 2v - 4

Nonplanarity of K5 and K3,3 follows immediately from this two

theorems. For K5,e = 10 > 9 = 3v – 6. Since K3,3 is triangle free, we

have e = 9 > 2v – 4. These graphs have too many edges to be planar.

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Another way of determining the planarity of graphs is through

the use of Euler’s formula. Euler's formula states that if a finite

connected planar graph is drawn in the plane without any edge

intersections, and v is the number of vertices, e is the number of

edges and r is the number of regions (area bounded by edges,

including the outer infinitely large area), then v − e + r = 2.

This study on the planarity of graphs will combine all these

important details and facts to make it possible for the readers to

understand and appreciate it in behalf of the complicated concepts it

contains.

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Chapter 3

METHODOLOGY

This thesis will make use pure descriptive research to determine

the different properties and other concepts related to graphs. The

research shall commence with the collation of preliminary concepts,

underlying principles and theoretical underpinnings which will serve as

the foundation of knowledge upon which pertinent analysis will be made.

Propositions will be made from observations and proven making

synthesized use of the collated concepts. Proving will be done in the most

simplified manner possible to eliminate confusion and complication on

the part of the readers who will consume this study either for its content

to understand and follow the process by which it was undertaken.

Examples and applications of planar graphs extracted from diverse

sources will be collated and presented still for purposes of simplification.

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Source of Data

In order to study the basic concepts on the Planarity of Graphs,

the researchers will refer to mathematics books, encyclopedia and

journals. Proofs to some theorems are also present in the internet but

the proofs are too complicated and very scholarly in approach. In order to

fulfill the objective to provide simplified proofs, more dependable sources

of information such as textbooks in mathematics and other reference

materials on Discrete Mathematics and Graph Theory will also be used.

Furthermore, other propositions and theorems will be proven by

the researchers in order to show the readers the necessary steps in

making proofs and instill to their minds the skills needed in proving.

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BIBLIOGRAPHY

Books

Balakrishnan, V. K. (1997).Shaum’s Outlines Graph Theory. New York:

The McGraw - Hill Companies, Inc.

Berge, Claude. (1958).The Theory of Graphs and its Applications. Great

Britain: Dunod.

Chartrand, Gary. (1977).Introductory Graph Theory. New York: Dover

Publications, Inc.

Rosen, Kenneth H. (2007).Discrete Mathematics and its Applications (6th

ed). New York: The McGraw - Hill Companies, Inc.

Trudeau, Richard J. (1993).Introduction to Graph Theory. The Kent State

University Press.

West, Douglas B. (2001).Introduction to Graph Theory (2nd ed). New

Jersey: Prentice – Hall, Inc.

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Online Journal Article

Dolev, D., Leighton T., & Trickey H. (1984). Planar embedding of planar

graphs. Advances in Computing Research, vol. 2, 147 – 161.

Retrieved January 10, 2011, from http://www.scansoft.com.

Online Encyclopedia Articles

Cycle (graph theory). (2010).Wikipedia, The Free Encyclopedia. Retrieved

December 11, 2010, from http://en.wikipedia.org/w/index.php?

title=Cycle_(graph_theory)&oldid=414207670.

Graph embedding. (2010).Wikipedia, The Free Encyclopedia. RetrievedDecember 11, 2010, from http://en.wikipedia.org/w/index.php?

title=Graph_embedding&oldid=415981619.

Graph isomorphism. (2010).Wikipedia, The Free Encyclopedia. Retrieved

January 1, 2011, from http://en.wikipedia.org/w/index.php?

title=Graph_isomorphism&oldid=418172522.

Graph theory. (2010).Wikipedia, The Free Encyclopedia. Retrieved

November 24, 2010, from http://en.wikipedia.org/w/index.php?

title=Graph_theory&oldid=418680102.

Graph (mathematics). (2010). Wikipedia, The Free Encyclopedia.

Retrieved, December 11, 2010, from http://en.wikipedia.org/w/

index.php?title=Graph_(mathematics)&oldid=414832159.

Inductive proof. (2009).The Unabridged Hutchinson Encyclopedia.

Retrieved December 11, 2010, from http://encyclopedia.

farlex.com/Inductive+proof.

Planar graph. (2010).Wikipedia, The Free Encyclopedia. Retrieved

November 24, 2010, from http://en.wikipedia.org/w/index.php?

title=Planar_graph&oldid=417969095.

Plane (geometry). (2010).Wikipedia, The Free Encyclopedia. Retrieved

December 11, 2010, from http://en.wikipedia.org/w/index.php?

title=Plane_(geometry)&oldid=419125080.

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Theorem. (2010). Wikipedia, The Free Encyclopedia. Retrieved November

24, 2010, from http://en.wikipedia.org/w/index.php?

title=Theorem&oldid=417693975.

Vertex (graph theory). (2010).Wikipedia, The Free Encyclopedia.

Retrieved, November 24, 2010 from http://en.wikipedia.org/w/

index.php?title=Vertex_(graph_theory)&oldid=395956133.

Weisstein, Eric W. (2010). Isomorphic Graphs.MathWorld -- A Wolfram

Web Resource. Retrieved December 11, 2010, from

http://mathworld.wolfram.com/IsomorphicGraphs.html.

Weisstein, Eric W. (2010). Subgraph.MathWorld -- A Wolfram Web

Resource. Retrieved December 11, 2010, from http://mathworld.

wolfram.com/Subgraph.html.

APPENDICES

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APPENDIX A

Letter to the Adviser

Don Mariano Marcos Memorial State University

South La Union Campus

College of Sciences

Agoo, La Union

Date:

Prof. Daisy Ann A. DisuDMMMSU – SLUC

College of Sciences

Agoo, La Union

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Ma’am;

We the undersigned BS Mathematics students are working on a thesis

entitled “On the Planarity of Graphs”. In this regard, we ask your

permission to be our adviser and help us further to formulate the

particular title of our study.

We are hoping for your positive response regarding this request.

Very truly yours,

Ralph Vincent E. Alambra

Ron Denny E. Bucasas

Approved by: ______________________

Prof. Daisy Ann A. Disu

APPENDIX B

Letter to the Panel Members

Don Mariano Marcos Memorial State University

South La Union Campus

College of Sciences

Agoo, La Union

Date:

Dr. Raquel D. Quiambao

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Dr. Delia V. Eisma

Mr. Tjaart Jan B. Estrada

Mr. Fernie V. Bucang

Sir/ Madam;

The undersigned students are conducting a research entitled “On the

Planarity of Graphs”.

In this connection, we request you to share your precious time in

evaluating this thesis. Your full cooperation and patience will surely

make this research successful. Your comments and suggestions in this

work would be greatly appreciated.

Thank you very much and more power.

Respectfully yours,

Ralph Vincent E. Alambra

Ron Denny E. Bucasas

APPENDIX C

THEOREMS

Theorem 1. If G is a plane graph, then the sum of the degrees of the

regions determined by G is 2e, where e is the number of

edges of G.

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Theorem 2. If a connected plane graph G has exactly v vertices, e edges,

and r regions, then v – e + r = 2.

Theorem 3. If G is planar and connected with e edges and v vertices

where v ≥ 3, then e ≤ 3v – 6.

Theorem 4. If G is planar and connected with e edges and v vertices

where v ≥ 3 and no circuits of length three, then e ≤ 2v – 4.

Theorem 5. K5 is nonplanar.

Theorem 6. K3,3 is nonplanar.

Theorem 7. Any subgraph of a planar graph is planar.

Theorem 8. Every supergraph of a nonplanar graph is nonplanar.

Theorem 9. Every expansion of K5 and K3,3is non planar.

Theorem 10. (Kuratowski’s Theorem) A graph is planar if and only if it

does not contain a subgraph which is homeomorphic to K5

and K3,3.

Theorem 11. If G is planar and H is isomorphic to G, then H is planar

also.

Theorem 12. If G is planar and connected then G has a vertex of degree

less than or equal to 5.

Theorem 13. A complete graph Kn is planar if and only if n ≤ 4.

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Theorem 14. A complete bipartite graph Km,nis planar if and only if m

≤ 2 or n ≤ 2.

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on the planarity of graphs 2

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