Planar Graphs. More precisely: there is a 1-1 function f : V R 2 and for each e E there exists a 1-1 continuous g e : [0, 1] R 2 such that

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1 Planar Graphs A graph G = (V, E) is planar i it can b drawn on th plan without dgs crossing xcpt at ndpoints a planar mbdding or plan graph. Mor prcisly: thr is a - unction : V R 2 and or ach E thr xists a - continuous g : [0, ] R 2 such that (a) = xy implis (x) = g (0) and (y) = g (). (b) implis that g (x) g (x ) or all x, x (0, ). g or its imag is rrrd to as a cur. c d Planar b a

2 Thorm (Fáry) A simpl planar graph has an mbdding in which all dgs ar straight lins. b d c a Not all graphs ar planar. Graphs can ha sral non-isomorphic mbddings. 2

3 Facs Gin a plan graph G, a ac is a maximal rgion S such that x, y S implis that x, y can b joind by a cur which dos not mt any dg o th mbdding Th abo mbdding has 7 acs. 0 is th outr or ininit ac. φ(g) is th numbr o acs o G. 3

4 Jordan Cur Thorm I is a - continuous map rom th circl S R 2 thn partitions R 2 \ (S ) into two disjoint connctd opn sts Int(), Ext(). Th ormr is boundd and th lattr is unboundd. As a consqunc, i x Int(), y Ext() and x, y ar joind by a closd cur C in R 2 thn C mts (S ). 4

5 K 5 is not planar. C is insid or outsid o C assum insid. 3 4 tc. din Jordan curs. C C 2 4 C Now no plac to put 5.g. i w plac 5 into C thn th 5 3 cur crosss th boundary o C. 5

6 Strographic Projction A graph is mbddabl in th plan i it is mbddabl on th surac o a sphr. z B a b A : R 2 S 2 \ {z}. (x, y) = ( 2x ρ, 2y ρ, ρ 2 ) ρ whr ρ = + x 2 + y 2. Gin an mbdding on th sphr w can choos z to b any point not an dg or rtx o th mbdding. Thus i is a rtx o a plan graph, G can b mbddd in th plan so that is on th outr ac. 6

7 Th boundary b() o ac o plan graph G is a closd clockwis walk around th dgs o th ac b( 0 ) = b( ) = b( 2 ) = 9 0 b( 3 ) = 7 7

8 Th dgr d() o ac is th numbr o dgs in b(). Each dg appars twic as an dg o a boundary and so i F is th st o acs o G, thn F d() = 2ɛ. A cut dg lik 6 appars twic in th boundary o a singl ac. 8

9 Dual Graphs Lt G b a plan graph. W din its dual G = (V, E ) as ollows: Thr is a rtx corrsponding to ach ac o G. Thr is an dg corrsponding to ach dg o G. and g ar joind by dg i dg is on th boundary o and g. Cut dgs yild loops. Thorm (a) G is planar. (b) G connctd implis G = G. 9

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11 Th ollowing is possibl: start with planar graph G and orm 2 distinct mbddings G, G 2. Th duals G, G 2 may not b isomorphic. G G 2 G has a ac o dgr 5 and so G has a rtx o dgr 5. G 2 has maximum dgr 4. Thus duality is a maningull notion w.r.t. plan graphs and not planar graphs.

12 φ(g) is th numbr o acs o plan graph G. (a) ν(g ) = φ(g). (b) ɛ(g ) = ɛ(g). (c) d G ( ) = d G (). Not that (c) says that th dgr o in G is qual to th siz o th boundary o in G. 2

13 Eulr s Formula Thorm 2 Lt G b a connctd plan graph. Thn ν ɛ + φ = 2. Proo By induction on ν. I ν = thn G is a collction o loops. φ = ɛ +. 3

14 I ν > thr must b an dg which is not a loop. Contract to gt G. G is connctd. But thn φ(g ) = φ(g) ν(g ) = ν(g) ɛ(g ) = ɛ(g) ν(g) φ(g) + ɛ(g) = ν(g ) φ(g ) + ɛ(g ) = 2 by induction. 4

15 Corollary All plan mbddings o a planar graph G ha th sam numbr ɛ ν + 2 acs. Corollary 2 I G is a simpl plan graph with ν 3 thn ɛ 3ν 6. Proo Ery ac has at last 3 dgs. Thus 2ɛ = d() 3φ. () F Thus by Eulr s ormula, ν ɛ ɛ 2. It ollows rom th abo proo that i ɛ = 3ν 6 thn thr is quality in () and so ry ac o G is a triangl. 5

16 Corollary 3 I G is a planar graph thn δ(g) 5. Proo νδ 2ɛ 6ν 2. Corollary 4 I G is a planar graph thn χ(g) 6. Proo Sinc ach subgraph H o G is planar w s that th colouring numbr δ (G) 5. Corollary 5 K 5 is non-planar. Proo ɛ(k 5 ) = 0 > 3ν(K 5 ) 6 = 9. 6

17 Corollary 6 K 3,3 is non-planar. Proo K 3,3 has no odd cycls and so i it could b mbddd in th plan, ry ac would b o siz at last 4. In which cas and so φ 4. 4φ F d() = 2ɛ = 8 But thn rom Eulr s ormula, 2 = φ, contradiction. 7

18 Kuratowski s Thorm A sub-diision o a graph G is on which is obtaind by rplacing dgs by (rtx disjoint) paths. Clarly, i G is planar thn any sub-diision o G is also planar. Thorm 3 A graph is non-planar i it contains a subdiision o K 3,3 or K 5. 8

19 Thorm 4 I G is planar thn χ(g) 5. By induction on ν. Triial or ν =. Suppos G has ν > rtics and th rsult is tru or all graphs with wr rtics. G has a rtx o dgr at most 5. H = G can b proprly 5-colourd, induction. I d G () 4 thn w can colour with a colour not usd by on o its nighbours. 9

20 Suppos d G () = 5. Tak som planar mbdding H = G can b 5-colourd. W can assum that c( i ) c( j ) or i j ls w can xtnd th colouring c to as priously. W can also assum that c( i ) = i or i 5. 20

21 Lt K i = {u V : c(u) = i or i 5 and lt H i,j = H[K i K j ] or i < j 5. First considr H,3. I and 3 blong to dirnt componnts C, C 3 o H,3 thn w can intrchang th colours and 3 in C to gt a nw propr colouring c o H with c ( ) = c ( 3 ) = 3 which can thn b xtndd to. So assum that thr is a path P,3 rom to 3 which only uss rtics rom K K 3. Assum w.l.o.g. that 2 is insid th cycl (,, 3, P,3, ), 2

22 2 P, Now considr H 2,4. W claim that 2 and 4 ar in dirnt componnts C 2, C 4, in which cas w can intrchang th colours 2 and 4 in C 2 to gt a nw colouring c with c ( 2 ) = c ( 4 ). I 2 and 4 ar in th sam componnt o H 2,4 thn thr is a path P 2,4 rom 2 to 4 which only uss rtics o colour 2 or 4. But this path would ha to cross P,3 which only uss rtics o colour and 3 contradiction. 22