Week 3: Connected Subgraphs

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1 Wk 3: Connctd Subgraphs Sptmbr 19, Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y Y. Th distanc btwn two vrtics x and y, dnotd d(x, y), is th minimal lngth of all xy-paths. If thr is no path btwn x and y, w dfin d(x, y) =. Tchniqu of Using Eignvalus: Thorm 1.1. Lt G b a simpl graph with n vrtics in which any two vrtics hav xactly on common nighbor. Thn G has a vrtx of dgr n 1. Consquntly, G must b obtaind from a family of disjoint triangls by gluing slctd vrtics, on from ach triangl, to a singl vrtx. Proof. Suppos it is not tru, i.., th maximal dgr (G) < n 1. W first show that G is rgular. Considr two non-adjacnt vrtics x and y. Lt f : N(x) N(y), whr f(v) is dfind as th uniqu common nighbor of v and y. W claim that f is injctiv. In fact, if f(u) = f(v) for distinct u, v N(x), thn f(u) is a common nighbor of u, v, y; now u and v hav two common nighbors x and f(u), a contradiction. Thus d(x) = N(x) N(y) = d(y). Likwis, d(y) d(x). So d(x) = d(y). This is quivalnt to say that any two adjacnt vrtics in Ḡ (th complmnt simpl graph of G) hav th sam dgr. W claim that G is rgular. To this nd, it suffics to show that Ḡ is connctd. Not that Ḡ has no isolatd vrtics, sinc th minimal dgr (Ḡ) = n 1 (G) > 0. Suppos Ḡ has two or mor connctd componnts. Tak two dgs i = u i v i from distinct componnts of Ḡ, i = 1, 2. Thn u 1 u 2 v 1 v 2 u 1 is a cycl of G. Thus u 1 and v 1 hav at last two common nighbors u 2, v 2, a contradiction. Lt G b k-rgular. Considr th numbr of paths of lngth 2 in G. Sinc any two vrtics hav xactly on common nighbor, thr ar ( n paths of lngth 2. For ach vrtx v, thr ar ( ( k paths with th middl vrtx v. It follows that n ( = n k. So n = k 2 k + 1. Lt A b th adjacncy matrix of G. Th (u, v)-ntry of A 2 is th numbr of (u, v)-walks of lngth 2. Thn A 2 has its diagonal ntris k and othr ntris 1. So A 2 = (k 1)I+J, whr 1

2 I is th idntity matrix and J is a matrix whos ntris ar 1. Not that J has ignvalu 0 with multiplicity n 1 and simpl ignvalu n. Sinc A 2 λi = (k 1 λ)i + J and n = k 2 k + 1, w s that A 2 has ignvalu k 1 with multiplicity n 1 and a simpl ignvalu k 2 with ignvctor (1,...,1) T. Sinc A 2 λ 2 I = (A λi)(a + λi), w s that A has th ignvalus ± k 1 with multiplicity n 1 and a simpl ignvalu k. Sinc th graph G is simpl, w hav tr(a) = 0 (th sum of its diagonal ntris). Rcall that th trac of A is th sum of its ignvalus countd with multiplicitis. W hav ±(n 1) k 1 + k = 0; it forcs that (n 1) k 1 = k. Th only possibl choic is that k = 2 and n = 3, i.., G is a triangl, whr (G) = 2. This is contradict to that (G) < n 1. Rmark: Th abov proof is intrsting, but th rsult is boring. 2 Eulr Tour A trail in a connctd graph is calld an Eulr tail if it travrss vry dg of th graph. Lt G b a connctd graph. A tour of G is a closd walk that travrss ach dg at last onc. An Eulr tour is a tour that travrss ach dg xactly onc. A graph is said to b Eulrian if it admits an Eulr tour. Thorm 2.1. A connctd graph G has an Eulr tour iff G has vn dgr at vry vrtx. Proof. Lt T = v 0 1 v n v n b an Eulr tour of G. Whn on travls along th Eulr tour and passs by a vrtx v, th prson must com towards v through on dg and dpart from v through anothr dg. Thn th numbr of tims coming towards v quals th numbr tims dparting from v. Thus th dgr of v must b vn. Considr a longst trail T = v 0 1 v n v n in G. W show that T is an Eulr tour. (a) Claim v 0 = v l. Suppos v 0 v l. Lt v l b appard k tims in th vrtx squnc (v 0, v 1,...,v l 1 ), say, v i1 = = v ik = v l, i 1 < < i ik < l. Thn th dgr of v l in T is 2k + 1. Sinc th dgr of v l in G is vn, thr is an dg in G but not in T incidnt with v l and anothr vrtx v. Thus T = Tv is a longr trail in G, a contradiction. (b) Claim that G E(T) has no dgs incidnt with a vrtx in T. Suppos thr is an dg E(G) E(T) incidnt with a vrtx v i V (T) and anothr vrtx v. Thn T = vv 1 i+1 v i+1 l v l (v 0 ) 1 v 1 v i 1 i v i is a longr trail in G, a contradiction. (c) Claim that T uss vry dg of G. Suppos thr xists an dg not usd in T. Lt b incidnt with vrtics u, v. Thn u, v V (T). Sinc G is connctd, thr is a shortst path P = u 0 x 1 u 1 x m u m from T to u, whr u 0 = u i and u m = u. W claim that x E(T). Suppos x 1 E(T), thn u 1 V (T); thus P = u 1 x 2 u 2 x m u m is a shortr path from T to u, a contradiction. Now sinc x 1 E(T), w hav a longr trail T = u 1 x 1 (u 0 )v i i+1 v i+1 l v l (v 0 ) 1 v 1 2 v i 1 i v i. 2

3 Again this is a contradiction. Sinc T is a closd trail that uss vry dg of G, w s that T is an Eulr tour. Corollary 2.2. A connctd graph G has an Eulr trail iff G has vn dgr at all vrtics or G has xactly two vrtics of odd dgr. Proof. Lt T := v 0 1 v 1 l v l b an Eulr trail of G. If v 0 = v l, thn T is an Eulr tour. By Thorm 2.1 G, G is an vn graph. If v 0 v l, w add a nw dg 0 btwn v 0 and v l. Thn T := T 0 v 0 is an Eulr tour of th graph G := G 0. Again by Thorm 2.1, G is an vn graph. It follows that G has xactly th two vrtics v 0, v l of odd dgr. It is Thorm 2.1 whn G is an vn graph. Lt G hav xactly two vrtics u and v of odd dgr. W add a nw dg btwn u and v to G to obtain a nw graph G. Thn G is a connctd vn graph. Thus G has an Eulr tour by Thorm 2.1. Rmov th dg from th Eulr tour for G, w obtain an Eulr trail for G. A cut dg of a graph G is an dg such that G has mor connctd componnts than G. Lmma 2.3. lt G b a connctd graph with a spcifid vrtx v. Assum that G is ithr an vn graph (formr cas) or G has xactly two vrtics u and v of odd dgr (lattr cas). (a) If d G (v) = 1 and is a link joining v to a vrtx w, thn G is th lattr cas. Morovr, G v is connctd, ithr having all vrtics of vn dgr or having xactly two vrtics v and w of odd dgr. (b) If d G (v) 2 and d G (v) is vn, thn G is th formr cas. Morovr, for ach dg joining v to a vrtx w, G is connctd, ithr having all vrtics of vn dgr or having xactly two vrtics v and w of odd dgr. (c) If d G (v) 2 and d G (v) is odd, thn G is th lattr cas. Morovr, thr xists an dg joining v to a vrtx w such that G is connctd, ithr having all vrtics of vn dgr or having xactly two vrtics v and w of odd dgr. Proof. (a) It is clar th lattr cas. If w = u, i.., d G (w) is odd, thn G v is a connctd vn graph. If w u, i.., d G (w) is vn, thn G v has xactly two vrtics u and w of odd dgr. (b) Sinc d G (v) is vn, it turns out that G is th formr cas. If is a loop at v, thn G hav th sam proprty as G. If is a link, thn G has xactly two odd-dgr vrtics v and w. W still nd to show that G is connctd. Suppos G has two connctd componnts G 1 and G 2 with v V (G 1 ) and w V (G 2 ). Thn G 1 has xactly on vrtx v of odd dgr. This is impossibl bcaus th numbr of odd-dgr vrtics is always vn. (c) It is clar that G is th lattr cas. If G has no cut dg at v, thn for any dg at v joining to a vrtx w, G is an vn graph if w = u or G has xactly two vrtics u and w of odd dgr. Lt G hav a cut dg at v. Thn G has two connctd componnts G 1 and G 2 with v V (G 1 ), d G1 (v) is vn, and d G (v) 3. If u V (G 1 ), thn G 1 has xactly on odd-dgr vrtx u; this is a contradiction. So u V (G 1 ), and G 1 is a connctd vn graph and d G1 (v) 2. Thn by (b) for any dg E(G 1 ) at v joining to a vrtx w, G 1 3

4 is connctd, ithr having all vrtics of vn dgr or having xactly two vrtics v and w of odd dgr. Consquntly, G is connctd, ithr having all vrtics of vn dgr or having xactly two vrtics u and w of odd dgr. Thorm 2.4. (Flury s Algorithm) Input: a connctd graph G = (V, E). Output: an Eulr tour, or an Eulr trail, or no Eulr trail for G. Stp 1 If thr ar vrtics of odd dgr, thn start at on such vrtx u. Othrwis, start at any vrtx u. St T := u and G := G. Stp 2 Lt v b th trminal vrtx of T. If thr is no dg rmaining at v in G, stop. (Now T is an Eulr tour if v = u and an Eulr trail if v u.) Stp 3 If thr is xactly on dg rmaining at v in G, joining v to anothr vrtx w, st T := Tw, G := G ; and rturn to Stp 2. Stp 4 If thr ar mor than on dg rmaining at v in G, choos on of ths dgs, say, an dg with nd-vrtics v to w, in such a way that G is still connct; T := Tw, G := G, and rturn to Stp 2. If such an dg can not b slctd, stop. (Thr is no Eulr trail.) Proof. Considr th pair (T, G ), calld an Eulrian pair of G, whr T is a trail of G, G is a connctd subgraph of G such that G is ithr an vn graph or has xactly two vrtics of odd dgr, th trminal vrtx v of T is a vrtx of G, and if G is th cas of having xactly two vrtics of odd dgr thn v is on of th two odd-dgr vrtics. Such a pair is said to b complmntary if E(G) = E(T) E(G ) (disjoint union). Th initial pair (T, G ) in Flury s algorithm is an Eulrian complmntary pair. Lt (T, G ) ban Eulrian complmntary pair with th trminal vrtx v of T in th procss of Flury s algorithm bfor ntring Stp 2. Now in Stp 2, if thr is no dg at v in G, thn E(G ) = (sinc G is connctd). It is clar that T is an Eulr trail of G. In Stp 3, w hav d G (v) = 1; thn by Lmma 2.3(c), (Tw, G ) is an Eulrian complmntary pair. In Stp 4, w hav d G (v) 2; thn by Lmma 2.3(b) and Lmma 2.3(c), (Tw, G ) is an Eulrian complmntary pair. Sinc all pair (T, G ) constructd in Flury s algorithm ar Eulrian complmntary pairs, and dgs of G ar rducing whn itrats, Flury s algorithm stops at (T, ) with T an Eulr train of G aftr finit numbr of itrats. v 8 v v v v v 7 v 6 v Figur 1: A graph with an Eulr trail. Exampl 2.1. An Eulr trail for th graph in Figur 1 is givn as v 3 3 v 4 4 v 5 5 v 6 6 v 7 7 v 8 8 v 1 1 v 2 2 v 3 12 v 9 14 v 4 16 v 6 15 v 9 10 v 2 9 v 8 11 v 9 13 v v 5

5 3 Connction in Digraphs A dirctd walk in a digraph D is an altrnating squnc of vrtics and arcs W := v 0 a 1 v 1...a l v l such that th arc a i has th tail v i 1 and had v i, i = 1,...,l. W call v 0 th initial vrtx and v l th trminal vrtx of W. Such a walk is rfrrd to a dirctd (v 0, v l )-walk; th subwalk of W from a vrtx v i to a vrtx v j is rfrrd to a (v i, v j )-sgmnt of W. A dirctd trail is a dirctd walk with distinct arcs. A dirctd path is a dirctd walk with distinct arcs and distinct vrtics, xcpt th possibl cas that th initial vrtx quals th trminal vrtx. Lt D b a digraph. A vrtx y is said to b rachabl from a vrtx x in D if thr is a dirctd (x, y)-path from x to y in D. Two vrtics x and y of D ar said to b strongly connctd if y is rachabl from x and x is rachabl from y in D. Strongly connctdnss is an quivalnc rlation on V (D). A sub-digraph of D inducd by an quivalnc class of th strong connctdnss is calld a strongly connctd componnt or strong componnt of D. A dirctd Eulr trail in a digraph D is a dirctd trail that uss vry arc of D. A closd dirctd Eulr trail is calld a dirctd Eulr tour. A digraph is said to b Eulrian if it has a dirctd Eulr tour. Thorm 3.1. Lt x and y b two vrtics of a digraph D. Thn y is rachabl from x in D iff (X, X c ) for vry subst X V (D) such that x X and y X. Proof. For ncssity, lt P b a dirctd path from x to y. For ach propr subst X V (D) with x X and y X c, th path P passs btwn X and X c. Th first arc of P from X to X c ; so (X, X c ). Convrsly, for sufficincy, suppos that y is not rachabl from x. Lt X b th st of vrtics rachabl from x. Thn y X c. Sinc vry vrtx of X c is not rachabl from x, thr is no arc from X to X c. So (X, X c ) = varnothing, a contradiction. Thorm 3.2. A connctd digraph D is Eulrian iff th in-dgr quals th out-dgr at vry vrtx of D. Proof. 4 Cycl Doubl Covr Cycl Covr, Cycl Doubl Covr A cycl covr of a graph G is a family F of subgraphs of G such that E(G) = E(H) and ach mmbr of F is a cycl. H F A cycl doubl covr of a graph G is a cycl covr such that ach dg of G blongs to xactly two mmbrs of F, i.., ach dg of G is covrd xactly twic by F. 5

6 Proposition 4.1. Lt G b a graph having a cycl covring C that ach dg of G is covrd at most twic. Thn G has a cycl doubl covr. Proof. Lt E 1 E(G) b th dg subst whos dgs ar covrd xactly on by C. Sinc C is a covring and ach dg of G is covrd at most twic by C, w s that G[E 1 ] is an Eulrian graph (i.. vn graph). So G[E 1 ] is a union of dg disjoint cycls, i.., G[E 1 ] has a covring C 1 that ach dg of G[E 1 ] is covrd xactly onc. Thus C 2 = C C 1 is a cycl doubl covring. of G. Cycl Doubl Covr Conjctur Conjctur 4.2. Evry graph (i.. having no cut dg) has a cycl doubl covr. 6