Directed acyclic graphs with the unique dipath property

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1 Direte yli grphs with the unique ipth property Jen-Clue Bermon, Mihel Cosnr, Stéphne érennes To ite this version: Jen-Clue Bermon, Mihel Cosnr, Stéphne érennes. Direte yli grphs with the unique ipth property. [Reserh Report] RR-69, 009, pp.9. <inri v> HAL I: inri Sumitte on 0 Aug 0 HAL is multi-isiplinry open ess rhive for the eposit n issemintion of sientifi reserh ouments, whether they re pulishe or not. The ouments my ome from tehing n reserh institutions in Frne or ro, or from puli or privte reserh enters. L rhive ouverte pluriisiplinire HAL, est estinée u épôt et à l iffusion e ouments sientifiques e niveu reherhe, puliés ou non, émnnt es étlissements enseignement et e reherhe frnçis ou étrngers, es lortoires pulis ou privés.

2 Direte yli grphs with the unique ipth property Jen Clue Bermon, Mihel Cosnr, Stéphne érennes MASCOTTE, joint projet CNRS-INRIA-UNSA, 004 Route es Luioles, B 9, F-0690 Sophi-Antipolis, Frne Astrt Let e fmily of ipths of DAG (Direte Ayli Grph) G. The lo of n r is the numer of ipths ontining this r. Let π(g, ) e the mximum of the lo of ll the rs n let w(g, ) e the minimum numer of wvelengths (olors) neee to olor the fmily of ipths in suh wy tht two ipths with the sme wvelength re r-isjoint. There exist DAGs suh tht the rtio etween w(g, ) n π(g, ) nnot e oune. An internl yle is n oriente yle suh tht ll the verties hve t lest one preeessor n one suessor in G (si otherwise every yle ontin neither soure nor sin of G). We prove tht, for ny fmily of ipths, w(g, ) = π(g, ) if n only if G is without internl yle. We lso onsier new lss of DAGs, whih is of interest in itself, those for whih there is t most one ipth from vertex to nother. We ll these igrphs U-DAGs. For these U-DAGs we show tht the lo is equl to the mximum size of lique of the onflit grph. We prove tht the rtio etween w(g, ) n π(g, ) nnot e oune ( result onjeture in n other rtile). For tht we introue goo lelings of the onflit grph ssoite to G n, nmely lelings of the eges suh tht for ny orere pir of verties (x, y) there o not exist two pths from x to y with inresing lels. Introution The prolem we onsier is motivte y routing, wvelength ssignment n grooming in optil networs. But it n e of interest for other pplitions in prllel omputing, where the grph will represent for exmple the preeene grph of progrm or for sheuling omplex opertions on pipeline opertors. A generi prolem in the esign of optil networs, [6, 9]), onsists of stisfying fmily of requests (or trffi mtrix) uner vrious onstrints lie pity onstrints. The optimiztion prolem ssoite onsists in esigning, for given fmily of requests, networ optimizing some riteri, suh s minimizing the numer of wvelengths or the numer of ADMs (A Drop Multiplexers). A request is stisfie y ssigning to it ipth in the networ. A fmily of requests is stisfie, if we n route them in suh wy tht the pity onstrints of the networ re stisfie. This is nown s the routing prolem. For given routing let us efine the lo of n r s the numer of routes (ipths) ontining it n the lo of the routing s the mximum lo of the rs. Typilly one wnts either to insure tht the lo of n r oes not exee the pity of this r or to minimize the lo of routing stisfying given fmily of requests. Mny one networs re now WDM optil ones. Inee wvelength ivision multiplexing (WDM) enles to use the nwith of n optil fier y iviing it in multiple non overlpping rtilly supporte y the CRC CORSO with Frne Teleom n y the Europen FET projet AEOLUS

3 frequenies or wvelength hnnels. Stisfying request in WDM optil networ onsists in ssigning to it route (ipth), ut lso wvelength, whih shll sty unhnge if no onversion is llowe. Therefore the onstrint is now tht two requests, hving the sme wvelength, hve to e route y two r isjoint ipths or, equivlently, two requests whose ssoite ipths shre n r, hve to e ssigne ifferent wvelengths. Hene the sre resoure is the numer of ville wvelengths. For given trffi mtrix, either one wnts to insure tht the fmily of requests n e stisfie with the ville numer of wvelengths or one wnts to minimize the numer of wvelengths use. This prolem is nown in the literture s the RWA (Routing n Wvelength Assignment) prolem. Note tht requests re stisfie on virtul (logil) networ whih is itself emee in the physil networ (in ft there might e mny lyers). It is the se for exmple when onsiering SONET/WDM rings or in MLS over WDM networs; in the ltter se the RWA prolem hs to e onsiere for the lightpths [9, 0]. Anywy, t the oneptul level of moeling of this rtile, the prolems re the sme n we will use the wor request to inite onnetion t the upper level. Minimizing the lo or/n the numer of wvelengths is iffiult prolem n in generl n Nhr prolem. These prolems hve een extensively stuie in the literture for vrious topologies or speil fmilies of requests lie multist or ll-to-ll (see for exmple the survey [] or [, 8]). Mny prtiulr ses where the minimum numer of wvelengths is equl to the minimum routing lo hve een given. For exmple, in [] it is shown tht for ny igrph n for multist instne (ll the requests hve the sme origin), there is equlity n oth prolems n e solve in polynomil time. For some topologies the lo might e esily ompute, ut the minimum numer of wvelengths is N-hr to ompute s it is relte to oloring prolems. This is the se for symmetri trees (see the survey [8]). However, for symmetri trees it hs een prove tht there is equlity for the ll to ll instne ([]) n pproximtion lgorithms hve een given ([8, ]). As the RWA prolem is very iffiult to solve, it is often split into two seprte prolems. First one solves the routing prolem y etermining ipths whih minimize the lo or re esy to ompute lie shortest pths. Then, the routing eing given, the wvelength ssignment prolem is solve. In tht se the input of the prolem is not fmily of requests ut fmily of ipths. We will enote y π(g, ) the mximum of the lo of ll the rs of the igrph G for the fmily. Determining the minimum numer w(g, ) of wvelengths (olors) neee to olor fmily of ipths in suh wy tht two ipths with the sme wvelength re r-isjoint is still N-hr in tht se. Inee it orrespons to fining the hromti numer of the onflit grph (lso lle the intersetion grph) ssoite to the igrph G n the fmily of ipths whose verties represent the ipths n where two verties re joine if the orresponing ipths re in onflit (tht is shre n r). There re exmples of topologiesn fmily of ipths where there re t most ipths using n r (π(g, ) = ), ut where we nee s mny wvelengths s we wnt. Figure shows the exmple for = 4 wvelengths) In the exmple we onsier ipths from s i to t i. The ipths strts in s i, then go lterntively right n own till they rrive t the ottom where they go right n up till they rrive t the estintion t i. Any two ipths interset so the onflit grph is omplete n we nee olors. However the lo of n r is t most. Therefore the rtio etween w(g, ) n π(g, ) is unoune in generl. Here we onsier the lss of Direte Ayli Grphs, DAGs, whih plys entrl role in rllel

4 n Distriute Computing. rt of our motivtion me when we trie to exten the results otine in [6] for pths motivte y grooming prolems for the pths ([4, 0]). In ft, we first prove tht for roote trees (irete trees where there is unique ipth from the root to ny vertex), for ny fmily of requests, the minimum numer of wvelengths is equl to the lo. The exmple given ove in Figure eing DAG there is no hope to oun rtio etween w(g, ) n π(g, ). In [5] we fully hrterize when w(g, ) = π(g, ) for DAG. In ft the neessry n suffiient onition is tht G oes not ontin wht we ll n internl yle, i.e. n oriente yle, suh tht ll the verties hve t lest one preeessor n one suessor in G (si otherwise ll yles ontin neither soure nor sin). Here, we give new shorter proof of this result eriving it from the se of trees where it is nown result s the onflit grph is perfet grph (see for exmple [5]). s t 4 s t s t s 4 t Figure : A pthologil exmple Then, we lso onsier new (to our est nowlege) lss of DAGs, whih is of interest in itself, those for whih there is t most one ipth from vertex to nother. Note tht, if the originl igrph hs the property tht for ny request (x, y) there is unique ipth from x to y, then it is equivlent to onsier fmily of requests or fmily of ipths. We ll this property the U (Unique ith roperty) n ll these igrphs U-DAGs. For these U-DAGs we show tht the lo is equl to the mximum size of lique of the onflit grph. In [5] we prove tht if n U-DAG hs only one internl yle, then for ny fmily of ipths w(g, ) = 4 π(g, ) n we exhiit n U-DAG n fmily of ipths rehing the oun. In this rtile we prove tht, for U-DAGs with lo, the rtio etween w(g, ) n π(g, ) nnot e oune ( result onjeture in [5]) For tht we introue the notion of goo lelings of the onflit grph ssoite to G n, nmely lelings of the eges suh tht for ny orere pir of verties (x, y) there o not exist two pths from x to y with inresing lels. We prove first tht, if G is n U-DAG with lo, then for ny fmily of ipths, the onflit grph C(G, ) hs goo leling. Then we lso show tht, if H is grph with goo leling, then there exists n U-DAG G with lo n fmily of ipths suh tht H = C(G, ). Finlly, we useg tht ft n proof of the existene of grphs with goo lelings ut hromti numer s lrge s we wnt to onlue.

5 ) ) Figure : An oriente yle () n n internl yle () Definitions We moel the networ y igrph G. The outegree of vertex x is the numer of rs with initil vertex x (tht is the numer of verties y suh tht (x, y) is n r of G). The inegree of vertex x is the numer of rs with terminl vertex x (tht is the numer of verties y suh tht (y, x) is n r of G). A soure is vertex with inegree 0 n sin vertex with outegree 0. A ipth is sequene of verties x, x,..., x suh tht (x i, x i+ ) is n r of G. If x = x the ipth is lle irete yle. A DAG (Direte Ayli Grph) is igrph with no irete yle. However the unerlying (unirete) grph otine y eleting the orienttion n hve yles. An (oriente) yle in DAG onsists therefore of n even sequene of ipths,,..., lternting in iretion (see Figure ). The verties insie the ipths hve inegree n outegree ; those where there is hnge of orienttion hve either inegree n outegree 0 or inegree 0 n outegree. An internl yle of DAG G is n oriente yle, suh tht ll its verties hve in G n inegree > 0 n n outegree > 0 ; si otherwise no vertex is soure or sin. Hene the verties where there is hnge of orienttion in the yle hve preeessor (resp. suessor) in G, if they re of inegree 0 (resp. outegree 0) in the yle (see Figure ). We will sy tht DAG hs the Unique th roperty if etween two verties there is t most one ipth. A igrph stisfying this property will e lle n U-DAG. If G is n U-DAG, then ny internl yle ontins t lest 4 verties where there is hnge of orienttion. Otherwise it woul onsist of ipth from x to y n reverse ipth from y to x n so there woul e two ipths from x to y. Finlly DAG with no yles is n oriente tree (its unerlying grph hs no yles n so is tree). Given igrph G n fmily of ipths, the lo of n r e is the numer of ipths of the fmily ontining e: lo(g,, e) = { : ; e } The lo of G for will e the mximum over ll the rs of G n will e enote y π(g, ). We will sy tht two ipths re in onflit (or interset) if they shre t lest one r. We will enote y w(g, ) the minimum numer of olors neee to olor the ipths of in suh wy tht two ipths in onflit (shring n r) hve ifferent olors. Note tht π(g, ) w(g, ).

6 The onflit grph (lso lle the intersetion grph) ssoite to the igrph G n the fmily of ipths hs s verties the ipths of, two verties eing joine if their ssoite ipths re in onflit (tht is interset = shre n r). It will e enote C(G, ). Then w(g, ) is the hromti numer γ of the onflit grph: tht is w(g, ) = γ(c(g, )) Note tht π is only upper oune y the lique numer of the onflit grph; inee the π ipths ontining n r e of mximum lo re pirwise in onflit.the following property shows tht if G is n U-DAG then π(g, ) is extly the lique numer ω(c(g, )) of the onflit grph. x y u v u v x y ) x y ) u u v u v x y v x y u v ) ) Figure : Helly property roperty If G is n U-DAG then the ipths in onflit hve the following Helly property : if set of ipths re pirwise in onflit, then their intersetion is ipth.therefore π(g, ) = ω(c(g, )) roof: If two ipths interset, then their intersetion is ipth. Inee suppose their intersetion ontins two ifferent ipths (x, y ) n (x, y ) in this orer. Then etween y n x there re two ipths, one vi n the other vi (see Figure ). So suppose n interset in only one intervl (x, y ), n intersets in n r isjoint intervl (u, v ). W.l.o.g. we my ssume tht v is efore x. Let intersets in the intervl (u, v ). Cse : v is efore u on. v nnot e fter y on otherwise there will e irete yle. So v is efore x on n we hve two ipths from v to x, one vi n the other one vi till u n then vi (see Figure ). Cse : u is fter v on. If u is efore x on we hve two ipths from v to x one vi

7 n the other going from v to u vi n to x vi. If u is fter y, we hve two ipths from v to u one vi n the other vi till y n (see Figure n ). Reltions etween π(g, ) n w(g, ) There exist DAGs G n set of ipths suh tht π(g, ) = n w(g, ) is s ig s we wnt (see Figure ). These DAGs hve mny internl yles. e e e G 5 pths e e Conflit grph Figure 4: Exmple for DAG with n internl yle In Figure 4, we give n exmple of DAG with one internl yle n set of 5 ipths suh tht π(g, ) = n w(g, ) =. The ipths re,, ;,, ;,, e ;,, e vi the seon ipth from to ;,, lso vi this seon ipth. The lo is n the onflit grph is yle of length 5 n so we nee olors to olor its verties. In ft s shown y the following theorem, if DAG G (whih n e n U-DAG) ontins n internl yle there nnot e equlity etween π(g, ) n w(g, ) for ll the set of ipths. Theorem If DAG G ontins n internl yle there exists set of ipths suh tht π(g, ) = n w(g, ) =. roof: Let us onsier n internl yle onsisting of ipths etween i n i n etween i n i (the inies re ten moulo ). So the i, i =,,...,, hve inegree 0 in the yle n the i, i =,,...,, hve outegree 0 in the yle. As the yle is internl, there exist verties i, i =,,..., joine to the i n verties i, i =,,..., to whih re joine the i. Let us te s set of ipths:,, ;,, ; i, i, i, i n i, i, i, i for i =,..., n,,,. The verties of the onflit grph ssoite to these ipths form yle of o length + in n so w = (see Figure 5). The exmple given in Theorem ove, for =, gives n U-DAG G with π = n set of 5 ipths suh tht the onflit grph is C 5 n therefore w =. Repling eh of these ipths with

8 Figure 5: Internl yle n fmily of ipths with π = n w =. h ientil ipths we otin fmily of 5h ipths with π = h n w = 5h giving rtio w π = 5 4. In ft the oun n e improve to 4 with the following exmple: Theorem There exists n U- DAG G with one internl yle n fmily of ipths suh tht 4 w(g, ) = π(g, ). roof: The following exmple is ue to Frééri Hvet (privte ommunition). It onsists of 8 ipths generting the onflit grph onsisting of yle of length 8 plus hors etween the ntipol verties (see Figure 6). Here gin π = n w = ; ut if we reple eh of these ipths with h ientil ipths we otin fmily of 8h ipths with π = h n w = 8h ; inee in the onflit grph n inepenent set hs t most verties n so we nee t lest 8h olors. Therefore this fmily stisfies the theorem (the reer n see the reltion with frtionl olouring). In [5] we showe tht the DAGs for whih for ny fmily of ipths, w(g, ) = π(g, ) re extly those with no internl yles. We give here simpler proof. Theorem 4 Let G e DAG. Then, for ny fmily of ipths, w(g, ) = π(g, ) if n only if G oes not ontin n internl yle. roof: Let G e DAG without internl yles. If G is n oriente tree, this is nown result s the onflit grph is perfet grph (see for exmple [5] or for polynomil lgorithm in O(n, ) [4]). If G is not n oriente tree, let G e the igrph otine s follows: reple eh soure s with + (s) neighors v i (i =,..., + (s)) y + (s) soures s i n join s i to v i. If ipth of ontins the r (s, v i ) ssoite in G the ipth otine y repling (s, v i ) y (s i, v i ). Do lso the sme trnsformtion for ll the sins repling the sin t with (t) neighors w j (j =,..., (s)) y (t) sins t j n (w j, t)

9 ' ' ' G ' ' ' ' ' ' ' ' ' Conflit grph Figure 6: An other U-DAG with π = n w =. y (w j, t j ). Let e the resulting fmily of ipths otine. G is n oriente tree ; inee, s there is no internl yle, ll the yles in G ontin either soure or sin. So we hve w(g, ) = π(g, ). By onstrution π(g, ) = π(g, ). To onlue let us olor the ipths of following the oloring of. If oes not ontin soure or sin it elongs to n we eep its olor. If it ontins (s, v i ) (resp (w j, t)) we give to it the olor of the ssoite pth in otine y repling (s, v i ) y (s i, v i ) (resp (w j, t) y (w j, t j )). We get vli oloring s there re no onflits etween two rs (s, v i ) n (s, v j ) (resp (w i, t) n (w j, t)). So we lso hve w(g, ) = w(g, ) n therefore w(g, ) = π(g, ). In [5] we se the question whether for ny U the rtio w π ws oune y some onstnt. We were only le to prove (with n involve proof) tht it ws the se when there ws only one internl yle otining rtio w π = 4 whih is the est possile in view of the exmple of theorem. We refer the reer to [5] for proof. Theorem 5, [5] Let G e n U-DAG with only one internl yle. Then for ny fmily of ipths n the oun is the est possile. w(g, ) 4 π(g, ) In the next setion we show tht the rtio is unoune t lest for U DAGs with lo. For tht we will hrterize their onflit grphs.

10 4 U DAGS with lo We will now show tht if G is n U-DAG with lo, for some fmily of ipths then its onflit grph H = C(G, ) mits goo leling of its eges. We give two equivlent efinitions of goo leling of the eges of grph H. Definition : Let us lel the eges of grph H with istint lels (for exmple the integers from to m = E(G) ). This leling is si to e goo if for ny orere pir of verties (x, y) there o not exist pths from x to y with inresing lels. Definition : Let us lel the eges of grph H with non neessrily istint lels. This leling is si to e goo if for ny orere pir of verties (x, y) there o not exist pths from x to y with non eresing lels. Clerly leling stisfying efinition stisfies efinition. Conversely let L e leling stisfying efinition n let the istint lels use in L e i i with < <... < n suppose i is repete λ i times. Let us efine the leling L s follows: we lel the λ i eges hving lel i in L with the istint lels i, i + ɛ i,..., i + (λ i )ɛ i where λ i ɛ i i+ i. Now onsier ny orere pir of verties (x, y). By efinition of L there exists t most one pth from x to y with non eresing lels. So ll the other pths ontin two onseutive eges with lels j n i suh tht j > i. Then in the leling L these two eges hve lso eresing lels s j j i+ > i + (λ i )ɛ j i. So L is leling with istint integers stisfying efinition. Let us give now some exmples of grphs hving goo leling or not: roperty 6 The yles C 4, C 5 n the onflit grph of Figure 6 with 8 verties hve goo leling roof: Let the C 4 e (,,,), then goo leling with efinition is otine y leling the eges {, } n {, } with lel n the eges {, } n {, } with lel. The leling L is otine y giving lel to {, }, to {, }, to {, } n 4 to {, }. Let the C 5 e (,,,,e), then goo leling with efinition is otine y leling the eges {, } {, } n {, e} with lel n the eges {, } n {, e} with lel 4. The leling L is otine y giving lel to {, }, to {, }, to {, e}, 4 to {, } n 5 to {, e}. For the onflit grph H of Figure 6 with 8 verties, goo leling with efinition is otine y leling the eges of the externl yle lterntively with lels n n the 4 igonls with lel. roperty 7 K, oes not mit goo leling. roof: This proof is ue to J-S Sereni. Let the verties of K, e respetively, n,, e. Suppose it mits goo leling L with efinition. Wlog we n suppose tht L(, ) < L(, ) < L(, e).

11 Then L(, ) > L(, ) otherwise there will e two inresing pths from to (,) n (,,,). Then L(, ) > L(, ) otherwise there will e two inresing pths from to (,,) n (,,). Then L(, e) > L(, e) otherwise there will e two inresing pths from to (,e,) n (,,). But we get ontrition s there re two inresing pths from to (,,) n (,e,). Theorem 8 Let G e n U-DAG with lo. Then for ny fmily of ipths, the onflit grph C(G, ) hs goo leling. roof: Rell (see the proof of property ) tht, if two ipths n Q interset, they interset in n intervl [x, y]. As the lo of G is, the rs of this intervl elong only to these ipths. Therefore, if G is the igrph otine from G y repling the intervl [x, y] y single r (x, y), then G hs the sme onflit grph H s G. The ege of H = C(G, ) joining the two verties n Q will orrespon to the intersetion intervl [x, y] of n Q in G tht is to the r (x, y) of G. (Note tht if in G we elete the rs with lo t most, tht is overe y t most one pth of, then there is one to one mpping etween the remining rs of G n the eges of the onflit grph H). Now we lel the rs of G oring to the topologil orer; tht is we lel the rs leving soure ; then we elete the rs lele getting igrph G n lel the rs leving soure in G n so on. As G n therefore G is DAG we n lel ll the rs of G. This inues leling of the eges of the onflit grph H, y giving to the ege joining the two verties n Q the lel of the r (x, y) of G ssoite to the intersetion intervl [x, y] of n Q. Let us now show tht it is goo leling of H. Consier non eresing pth in H, from to Q, =,,..., = Q n let (x i, y i ) e the r of G ssoite to the intersetion of i n i+. As the lels re non eresing, then (x i, y i ) is in the topologil orer efore (x i+, y i+ ) n so y i is efore x i+ in i+. So this non eresing pth in H inues in G ipth x, y, x, y,..., x, y (in ft tht implies tht the lels re stritly inresing). Suppose we hve two non eresing pths in H from vertex to vertex Q, then we hve in G two ipths x, y, x, y,..., x, y n x, y, x, y,...,, x m, y m with x, y n x, y elonging to n x, y n x m, y m elonging to Q. Wlog we n suppose x is fter y on. If x m is fter y on Q, we hve two ipths joining y n x m nmely y, x, y,,..., x, y, x m n y, x, y, x, y,...,, x m, y m, x m. If y m is efore x on Q, we hve two ipths joining y n x nmely y, x, y,,..., x n y, x, y, x, y,...,, x m, y m, x. Therefore G n so G nnot e n U igrph. So using the property 7 we otin the following orollry prove in [5]: Corollry 9 Let G e n U-DAG with lo. Then its onflit grph nnot ontin K,. Theorem 0 Let H e grph with goo leling. Then there exists n U-DAG G with lo n fmily of ipths suh tht H = C(G, ). roof: To eh ege {, Q} in H let us ssoite in G two verties x Q n y Q joine y the r (x Q, y Q ). Now for eh vertex in H, orer its neighors Q, Q,..., Q h oring to the lels of the

12 eges {, Q i } tht is L(, Q ) < L(, Q ) <... < L(, Q h ). Then, for i =,,..., h, let us ientify the vertex y Qi to x Qi+ ). To the vertex in H we ssoite in G the ipth = (x Q, y Q = x Q, y Q,..., x Qh, y Qh ). The fmily of ipths onsists of the ipths ssoite to eh vertex of H. The onflit grph ssoite to the grph G n the fmily of ipths is extly the grph H. G hs lo s n r x Q, y Q of G elongs extly to the two ipths n Q. G is n U s ipth in G orrespons to n inresing pth in H n so if there were two ipths in G joining some y Q to x Q there will e two inresing pths in H from to. If we pply the onstrution of the proof to the grph H with verties of Figure 6 we get extly the grph G n the ipths of the exmple. Remr: there is no equivlene etween the two properties G eing U with lo n its onflit grph hving goo leling. Inee there exist igrphs whih re not U ut whose onflit grph hs goo leling : for exmple onsier the grph of Figure 4 with the ipths,, ;,, ;,, e ; n,,, e vi the seon ipth from to ; it hs C 4 s onflit grph. Theorem There exists fmily of grphs with goo leling n hromti numer s lrge s we wnt. roof: Consier regulr grph H of egree t most, girth > + n with lrge hromti numer. The existene of suh grphs hs een shown in [7, 7]. The eges of H n e prtitione in t most + mthings (olortion of the eges of grph with t most + olors y Vizing s theorem). Let us give to the eges of eh mthing ifferent lel,,..., +. Then ny non eresing (in ft inresing s there nnot e two onseutive eges with the sme lel) pth in H hs t most + eges. Therefore there nnot exist two inresing pths otherwise there will e yle in H of length + ontriting the vlue of the girth. Now using theorems 0 n we re le to nswer the question se in [5]. Theorem There exist U- DAGSsn fmily of ipths with lo π(g, ) = n w(g, ) s lrge s we wnt. 5 Conlusions Mny questions re worth of eing investigte. In this rtile we stuy the reltions etween the mximum of the lo π(g, ) of ll the rs n w(g, ) the minimum numer of wvelengths in prtiulr for the lss of U-DAGs. In prtiulr we hve the following onjeture ( speiliztion of prolem se in []). CONJECTURE : If G is DAG n if we onsier the ll to ll fmily of ipths (i.e for eh ouple (x,y) onnete y ipth we hve request. Then w(g, AT A) = π(g, AT A). In se of inry trees we hve otine n expliit formul for oloring them in tht se.

13 A nturl question is to see when property is vli. QUESTION : When is π(g, ) = ω(c(g, )). QUESTION : Given n unirete grph when is it possile to orient its eges suht tht the igrph otine is U. Tht is n N-Hr prolem ut lsses of suh grphs oul e exhiite. For grphs with one or smll numer of internl yles we hve the following questions. QUESTION 4 Is the Theorem 5 true lso for DAGS (not neessrily U) with extly one internl yle? QUESTION 5: is there simple proof of the Theorem 5 with one internl yle? QUESTION 6: Wht is the oun if we hve extly internl yles? In preliminry version of this pper we se for hrteriztion of grphs with goo leling. In prtiulr we se if it ws polynomil or not to eie if grph hs goo leling. In following pper [], the uthors exhiit infinite fmilies of grphs for whih no suh ege-lelling n e foun. They lso show tht eiing if grph mits goo ege-lelling is N-omplete. Finlly, they give lrge lsses of grphs mitting goo ege-lelling lie forests, C -free outerplnr grphs, plnr grphs of girth t lest 6. A lst question onsists in extening the results to U-DAGS with lo > perhps y using hypergrphs. QUESTION 7 Chrterize U-DAGS with lo or lo h. Anowlegement: We thn J. Arujo, D.Couert n F.eix for their helpful remrs. Referenes [] J. Arujo, N. Cohen, F. Giroire, n F. Hvet. Goo ege-lelling of grphs. In proeeings of the Ltin-Amerin Algorithms, Grphs n Optimiztion Symposium (LAGOS 09), Eletroni Notes in Disrete Mthemtis, Grmo, Brzil, Novemer 009. Springer. [] B. Beuquier, J-C. Bermon, L. Grgno,. Hell, S. érennes, n U. Vro. Grph prolems rising from wvelength-routing in ll-optil networs. In IEEE Worshop on Optis n Computer Siene, Genev, Switzerln, April 997. [] B. Beuquier,. Hell, n S. érennes. Optiml wvelength-route multisting. Disrete Applie Mthemtis, 84:5 0, 998.

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