4–Solvingproblems Remainingtime 3–Quicklesson 20mins 2–Thesolutions 10mins Previousperiod 1–Appetizer:threeeulerianproblems • Materials: • • Objectives: • Euleriangraphs Season2Episode19Timeframe2periods

Eulerian graphs

Season 2

Episode 19 Time frame 2 periods

Objectives :

•

^{Introdue the voabularyabout graphs.}

•

^{Disover the main results about eulerianand} traversablegraphs.

Materials :

•

^{Lesson (36 opies).}

•

^{Problems(36 opies).}

1 – Appetizer : three eulerian problems Previous period

Three famous problems (The bridgesof Konigsberg and the house with 5rooms, with or

withoutadoortotheoutside)are presented tothestudents.They havetosolvethem for

the next period.

2 – The solutions 10 mins

The solutions tothe three problems are given by the teaher or students.

3 – Quick lesson 20 mins

The voabulary about graphs is introdued, and then the main results about eulerian

graphs.

4 – Solving problems Remaining time

Students are working in pairs. They have to solve 10 problems about eulerian graphs.

Eah problemis worth 2pointsand amarkisgiven toeahpair atthe endof the seond

Document Lesson

Agraph

G

isasetofpoint,the verties,alongwithaset oflinesegmentsorurvesjoining some of these verties, the edges. The order of a graphis its number of verties while its

size is itsnumberof edges.

1 Conneted graphs

Let

u

and

v

be two verties in a graph

G

. As an example, we will onsider the graph drawn below.

The verties

u

and

v

are said to be adjaent if the edge

uv

is part of the graph. In the same situation,we say that the edge

uv

isinident to the verties

u

and

v

.

The degree of a vertex is the number of edges adjaent to it. If the degree of a vertex is

even, we simplysay that it'saneven vertex, and similarlyfor anodd vertex.

•

^A

u − v

walk isasequene of adjaent verties beginningwith

u

and endingwith

v

. In the example, a

v ₁ − v ₃

walkis

v ₁

,

v ₂

,

v ₃

,

v ₇

,

v ₂

,

v ₃

,

v ₄

,

v ₃

.

•

^A

u − v

trail isa

u − v

walkwhihdoesnot repeat any edge. Inthe example, a

v ₁ − v ₃

trailis

v ₁

,

v ₂

,

v ₇

,

v ₃

,

v ₄

,

v ₃

.

•

^A

u − v

path isa

u − v

trailwhihdoesnotrepeat anyvertex.In theexample,a

v ₁ − v ₃

path is

v ₁

,

v ₂

,

v ₇

,

v ₃

.

Twoverties

u

and

v

are onneted if either

u = v

orthere exists a

u − v

path from

u

to

v

.A graphisonneted if every twoverties are onneted.Otherwise,it isdisonneted.

Our example is obviouslya onneted graph.

•

^A

u − v

trail whih ontains at least three verties and suh that

u = v

is alled a iruit. In our example,

v ₂

,

v ₇

,

v ₃

,

v ₆

,

v ₅

,

v ₄

,

v ₃

,

v ₂

is airuit.

•

^{A iruit whih does not repeat any verties is alled a yle. In our example,}

v ₂

,

v ₇

,

v ₃

,

v ₂

is ayle.

2 Eulerian graphs

Airuitontainingalledgesofagraph

G

isalledaneulerianiruit.Agraphontaining an eulerian iuit is alled aneulerian graph. It's easy to see that our example is not an

eulerian graph,beause of the edge

v ₁ v ₂

.

Theorem 1

A graph is eulerian if and only if it is connected and every vertex is even.

If agraph

G

has a trail, not a iruit,ontainingall edges of

G

,

G

is alleda traversable graph and the trailis alled aneulerian trail.

Wean see, by trialand error, that our exampleis not a traversable graph.

Theorem 2

A graph is traversable if and only if it is connected and has exactly two odd vertices.

Furthermore, any eulerian trail in such a graph begins at one of the odd vertices and

ends at the other.

Document Problems

Problem 1

Classify the following graphs as eulerian, traversable or neither. For the rst two kinds,

nd an eulerianiruit ortrail.

Problem 2

Suppose you hold a summer job as a highway inspetor. Among your responsabilities,

you must periodiallydrive along the several highways shown shematiallyin the gure

below and inspet the roads for debris and possible repairs. If you live in town A, is it

possibletondaroundtrip,beginningandendingatA,whihtakesyouovereahsetion

ofhighwayexatlyone? Ifyouwere tomovetotownB, woulditbepossibletondsuh

Problem 3

Thegurebelowshows theblueprintofahouse.Canapersonwalkthrougheahdoorway

of this houseone and only one? If so,show howit an be done.

Problem 4

A letter arrier is responsible for delivering mail to houses on both sides of the streets

shown in the gure below. If the letter arrier does not keep rossing s street bak and

forth toget to houses onboth sides of astreet, itwill beneessary for her to walk along

a street atleast twie, one on eah side, to deliver the mail.Is it possible for the letter

arrier to onstrut a round trip so that she walks on eah side of every street exatly

Problem 5

Givean exampleof a graph ororder 10 whih is

1

^{. eulerian;}

2

^. traversable;

3

^{. neither euleriannor} traversable.

Problem 6

A omplete graph of order

n

is a graph with

n

verties suh that every two verties are onneted by anedge.

1

^{. Drawthe omplete graphsof order 2, 3,4, 5,6.}

2

^{. Find out whihones of these graphs are} traversable oreulerian.

3

^{. Find asimple rule to deide whena omplete graph is}traversable oreulerian.

Problem 7

Anypolyhedronan beturnedintoaplanargraph(withnointerseting edges)bysimply

reproduing its verties and edges on a plane. Some edges may have to be drawn as

non-straightlines inthe proess.

1

^{. Drawthe graphsof a} tetrahedron,a ubeand an otahedron.

2

^{. Find out whihones of these three graphs are} traversableor eulerian.

Problem 8

Prove that if a graph istraversable, aneulerian graph an be onstruted fromit by the

addition ofa single edge.

Problem 9

Is the previous property alsotrue if wereplae the word addition by deletion?

Problem 10

Weknow thata onnetedgraph with noodd vertiesontains aneulerianiruit,and a

onnetedgraphwithexatlytwooddvertiesontainseneuleriantrail.Trytodetermine

what speial property is exhibited by a onneted graph with exatly four odd verties.

Tryto prove your answer.