On Gallai s and Hajós Conjectures for graphs with treewidth at most 3

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1 On Glli s nd Hjós Conjetres for grphs with treewidth t most 3 rxi: [mth.co] 14 Jn 2017 F. Botler 1 M. Sminelli 2 R. S. Coelho 3 O. Lee 2 1 Fltd de Cienis Físis Mtemtis Uniersidd de Chile 2 Institto de Comptção Uniersidde Estdl de Cmpins 3 Institto Federl do Norte de Mins Geris Noemer 12, 2018 Astrt A pth (resp. le) deomposition of grph G is set of edge-disjoint pths (resp. les) of G tht oers the edge set of G. Glli (1966) onjetred tht eer grph on n erties dmits pth deomposition of size t most (n+1)/2, nd Hjós (1968) onjetred tht eer Elerin grph on n erties dmits le deomposition of size t most (n 1)/2. Glli s Conjetre ws erified for mn lsses of grphs. In prtilr, Loász (1968) erified this onjetre for grphs with t most one erte of een degree, nd Per (1996) erified it for grphs in whih eer le ontins erte of odd degree. Hjós Conjetre, on the other hnd, ws erified onl for grphs with mimm degree 4 nd for plnr grphs. In this pper, we erif Glli s nd Hjós Conjetres for grphs with treewidth t most 3. Moreoer, we show tht the onl grphs with treewidth t most 3 tht do not dmit pth deomposition of size t most n/2 re isomorphi to K 3 or K 5 e. Finll, we se the tehniqe deeloped in this pper to present new proofs for Glli s nd Hjós Conjetres for grphs with mimm degree t most 4, nd for plnr grphs with girth t lest 6. 1 Introdtion In this pper, ll grphs onsidered re simple, i.e., ontin no loops or mltiple edges. A deomposition D of grph G is set {H 1,...,H k } of edge-disjoint sgrphs of G tht This reserh hs een prtill spported CNPq Projets (Pro / nd /2014-7), Fpesp Projet (Pro. 2013/ ). F. Botler is prtill spported CAPES (Pro ), Millenim Nles Informtion nd Coordintion in Networks (ICM/FIC RC ), nd CONICYT/FONDECYT/POSTDOCTORADO M. Sminelli is spported CNPq (Pro /2016-6). O. Lee is spported CNPq (Pro / nd /2012-5). e-mils: fotler@ime.sp.r (F. Botler), msminelli@i.nimp.r (M. Sminelli), roelho@ime.sp.r (R. S. Coelho), lee@i.nimp.r (O. Lee) 1

2 oer the edge set of G. We s tht D is pth (resp. le) deomposition if H i is pth (resp. le) for i = 1,...,k. We s tht pth(resp. le) deomposition D of grph (resp. n Elerin grph) G is minimm if for n pth (resp. le) deomposition D of G we he D D. The size of minimm pth (resp. le) deomposition is lled the pth nmer (resp. le nmer) of G, nd is denoted pn(g) (resp. n(g)). In this pper, we fos in the following onjetres onerning minimm pth nd le nmers of grphs (see [4, 17]). Conjetre 1 (Glli, 1966). If G is onneted grph, then pn(g) V(G) Conjetre 2 (Hjós, 1968). If G is n Elerin grph, then n(g) V(G) 1 2. In 1968, Loász [17] proed tht grph with n erties n e deomposed into t most n/2 pths nd les. The net theorem is direted onseqene of this reslt. Theorem 1.1 (Loász, 1968). If G is grph with n erties nd ontins t most one erte of een degree, then pn(g) = n/2. Per [18] etended Theorem 1.1 s follows. Theorem 1.2 (Per, 1996). If G is grph with n erties in whih eer le ontins erte of odd degree, then pn(g) n/2. In 2005, Fn [11] etended Theorem 1.1 een more, t Conjetre 1 is still open. Reentl, one of the thors [5] erified Conjetre 1 for fmil of een reglr grphs with high girth ondition, nd Jiménez nd Wkshi [16] erified it for fmil of tringle-free grphs. For more reslts onerning Conjetre 1, we refer the reder to [8, 10, 12, 13]. Althogh these onjetres seems er similr, Conjetre 2 ws onl erified for grphs with mimm degree 4 [14] nd for plnr grphs [19]. The tehniqe presented in this pper showed to e sefl to del with oth Conjetres 1 nd 2. Let G e onter-emple for Conjetre 1 with minimm nmer of erties. Or tehniqe onsists in finding sgrph H of G sh tht, for some positie integer r, thegrphg = G E(H) ontinstmost V(G) 2r non-isoltederties, nd pn(h) r. Moreoer, weshowhowtootinh inshwthtforeeromponentc of G we he pn(c ) V(C ) /2. Therefore, the deomposition D of G otined joining minimm pth deompositions of H nd G is sh tht D V(G) /2. The grph H is lled n r-reding sgrph nd is disssed in Setion 2. As prodt of or min reslt (Theorem 3.2), the onl grphs with treewidth t most 3 nd pth nmer etl (n+1)/2 re isomorphi to K 3 nd K 5 (the grph otined from K 5 remoing etl one edge). For Conjetre 2 the proedre is nlogos. The min ontritions of this pper re the following. We erif Conjetre 1 for grphs with treewidth t most 3, grphs with mimm degree t most 4, nd plnr grphs with girth t lest 6. In ft, for ll these ses, we proe the Conjetre 3, whih is strengthening of Conjetre 1. Also, we erif Conjetre 2 for grphs with treewidth t most 3 nd present new proof for the se of grphs with mimm degree t most 4. Conjetre 3. Let G e onneted grph with n erties. If E(G) (n 1) n/2, then pn(g) n/2. Otherwise, pn(g) = n/2. 2

3 Etended strts of prts of this work [6, 7] were epted to LAGOS 2017 nd to the Brzilin Compter Soiet Conferene (CSBC 2017). While writing this pper, we lerned tht Bonm nd Perrett [3] erified Conjetre 1 for grphs with mimm degree 5. Howeer, sine the dne of the stte-of-the-rt of Conjetre 1 in this diretion is er reent, we eliee tht oth tehniqes nd proofs re importnt to the litertre. This work is orgnized s follows. In Setion 2, we define reding sgrphs, present some tehnil lemms, nd onfirm Conjetre 1 for plnr grphs with girth t lest 6. In Setion 3, we settle Conjetres 1 nd 2 for grphs with treewidth t most 3 nd, in Setion 4, we present new proofs for Conjetres 1 nd 2 for grphs with mimm degree t most 4. Finll, in Setion 5, we gie some onlding remrks. Nottion The si terminolog nd nottion sed in this pper re stndrd (see, e.g. [9]). All grphsonsideredhererefinitendhenoloopsnormltipleedges. LetG = (V,E)e grph. ApthP ingisseqene ofdistintertiesp = 0 1 l shtht i i+1 E. It is lso onenient to refer to pth P = 0 1 l s the sgrph of G inded the edges i i+1, for i = 1,...,l 1. For ese of nottion, gien n edge, we denote G+ the grph (V {,},E {}), nd G the grph (V,E \{}). If E is set of edges, then G+E (resp. G E ) denotes G+ e E e (resp. G e E e). Gien two (not neessril disjoint) grphs G nd H, we se G+H to denote the grph ( V(G) V(H),E(G) E(H) ). We write G1 G 2 (resp. G 1 G 2 ) to denote tht G 1 is isomorphi (resp. non-isomorphi) to G 2. Gien set U nd n element e, we define U +e = U {e} nd U e = U\{e}. Inthispper, wefreqentlonttheisoltedertiesv i ingrphgftertheremol of the edges of sgrph H G. To oid the introdtion of more nottion, when ler from ontet, the grph G E(H) denotes either the grph ( V(G),E(G) E(H) ) or the grph ( V(G) V i,e(g) E(H) ). Let G e onneted grph. We s tht set S E(G) of edges of G is n edge seprtor if G S is disonneted. If S is miniml edge seprtor, i.e., S is n edge seprtor, t S is not n edge seprtor for eer S S with S S, we s tht S is n edge-t. The figres in this pper re depited s follows. Solid edges nd fll erties illstrte edges nd erties tht re present in the grph, while dshed edges nd empt erties illstrte edges nd erties tht m e in the grph, nd loosel dotted edges illstrte edges tht re not present in the grph. Stright edges illstrte simple edges, while snke edges illstrte pths with possile internl erties. d Figre 1: the erties,, d illstrte fll/present erties; the erte illstrtes n empt/possile erte; the edge d illstrtes solid/present edge; the edges,, nd d illstrte dshed/possile edges; the edge illstrtes loosel dotted/non-present edge; the edge d illstrtes snke edge/pth. 3

4 2 Reding sgrphs In this setion, we define reding sgrphs nd present some reslts tht llow s to del with them (see Lemm 2.7). Let G e grph nd let H e sgrph of G. Gien positie integer r, we s tht H is n r-reding sgrph of G if G E(H) hs t lest 2r isolted erties nd pn(h) r. If H is 1-reding sgrph of G, then we s tht H is reding pth of G. We s tht H is reding sgrph of G if H is n r-reding sgrph of G for some positie integer r. We s tht grph G with n non-isolted erties is Glli grph if pn(g) n/2. Note tht, in this se, G is lso n n/2 -reding sgrph of itself. The net lemm formlizes the reltion etween Glli grphs nd r-reding sgrphs. We lso osere tht sine pn(k 3 ) = 2 nd pn(k 5 ) = pn(k5 ) = 3, the grphs K 3, K 5 nd K5 re not Glli grphs. Lemm 2.1. Let G e grph, nd let H G e reding sgrph of G. If G E(H) is Glli grph, then G is Glli grph. Proof. Let G nd H e s in the sttement, where H is n r-reding sgrph of G. B the definition of r-reding sgrph, G E(H) hs t lest 2r isolted erties, nd sine G E(H) is Glli grph, pn(g E(H)) (n 2r)/2 = n/2 r. Ths, there is pth deomposition D of G E(H) with size t most n/2 r. Sine H is n r-reding sgrph, there is pth deomposition D H of H with size t most r, nd hene D D H is pth deomposition of G with size t most n/2. NowwereletoerifConjetre 1forplnrgrphswithgirthtlest 6. Fortht, we se the ft tht eer onneted plnr grph with n erties nd girth t lest 6, ontins t lest three erties of degree t most 2. Indeed, let G e onneted plnr grphwithgirtht lest 6, ndlet n ethe nmer oferties of G withdegree t most 2. Sine eer erte of G hs degree t lest 1, if n 2, then 2 E(G) = V (G) d() n +3(n n ) = 3n 2n 3n 4. On the other hnd, sine G hs girth t lest 6, the ondr wlk of eh fe of G ontins t lest 6 edges. Ths, 2 E(G) 6f, where f is the nmer of fes of G. B Eler s forml, we he n+f E(G) = 2, whih implies 6 = 3n+3f 3 E(G) 2 E(G) +4+ E(G) 3 E(G) = 4, ontrdition. Theorem 2.2. Eer onneted plnr grph with girth t lest 6 is Glli grph. Proof. Sppose, for ontrdition, tht the sttement does not hold, nd let G e onter-emple for the sttement with minimm nmer of erties. As osered oe, G ontins t lest three erties of degree t most 2. Let P e shortest pth in G joining two of these erties, s nd, nd let S e the set of edges of G E(P ) inident to or. Pt P = P + S. Sine nd he degree t most 2, S ontins t most one edge inident to eh nd, nd sine G hs girth t lest 6, P is pth. Moreoer, nd re isolted in G = G E(P) nd hene P is reding pth. Note tht eh omponent of G is plnr grph with girth t lest 6. B the minimlit of G, eh omponent of G is Glli grph, hene G is Glli grph. Therefore, Lemm 2.1, G is Glli grph, ontrdition. Now let G e Glli grph on n non-isolted erties nd let D e minimm pth deomposition of G. Sine eh pth in G ontins t most n 1, we he E(G) = 4

5 P D E(P) P D (n 1) n/2 (n 1). Note tht this ineqlit holds wheneer n is een. In the se where n is odd, we s tht grph G with n erties is qsiomplete (or n odd semi-liqe see [3]) if E(G) > n/2 (n 1). Therefore, it is ler tht no qsi-omplete grph is Glli grph. The qsi-omplete grphs re preisel the grphs otined from K n (with n odd) remoing t most n/2 1 edges. A diret implition of or min reslt is tht the onl non-glli prtil 3-tree re the qsi-omplete prtil 3-trees, i.e., the omplete grph K 3 nd K5. Or min tehniqe onsist in finding reding sgrphs of gien grphs. The following reslts llow s to onstrt r -reding sgrphs from r-reding sgrphs nd grphs isomorphi to K 3, K 5, nd K5. Note tht the following reslts reqire onl the grph G to e onneted nd need no other propert s, for emple, eing prtil 3-tree. The proof of the net lemm follows the proof of Lemm 3.2 in [5]. Gien le C in grph G, hord of C is n edge in G E(C) tht joins two distint erties of C. Gien pth P = 0 1 k, if i < j, we denote P( i, j ) the spth i i+1 j. Lemm 2.3. Let G e onneted grph tht dmits deomposition into pth P nd le C. (i) if P ontins t most one hord of C, then G dmits deomposition into two pths P 1 nd P 2 sh tht P 1 ontins etl one edge of C; nd (ii) if C hs length t most 5 nd P ontins t most three hords of C, then pn(g) = 2. Proof. Let G, P, nd C e s in the sttement. Let P = 0 1 k nd C = 0 l 0. Let z 0,...,z s e the erties in V(C) V(P) in the order tht the pper in 0 k, nd sppose withot loss of generlit tht z 0 = 0. Clim 1. If { 1, l } V(P) or P(z i 1,z i ) hs length t lest 2 for some z i { 1, l }, then G n e deomposed into two pths P 1 nd P 2 sh tht P 1 ontins etl one edge of C. Proof. If there eists erte { 1, l } sh tht / V(P), then P 1 = P E(P( 0,z 0 )) + z 0 nd P 2 = G E(P 1 ) deompose G s desired (see Figre 2). If P(z i 1,z i ) hs length t lest 2 for some z i { 1, l }, then there is neighor z of z i in P(z i 1,z i ). Note tht z is not erte of C. Ths, P 1 = P E(P( 0,z 0 )) z z i +z 0 nd P 2 = G E(P 1 ) deompose G s desired (see Figre 2). k k k k z z 0 0 z 0 z z z 0 0 z i 0 z i () () Figre 2: The illstrtion on the left of figres () nd () shows the pth P nd the le C in red nd lk, respetiel, while the illstrtion on the right shows the pths P 1 nd P 2 in red nd lk, respetiel. 5

6 Now we proe item (i). Sppose tht P ontins t most one hord of C. If { 1, l } V(P), then the reslt follows Clim 1. Ths, let {z i,z j } = { 1, l }. Sine P ontins t most one hord of C, t lest one etween P(z i 1,z i ) or P(z j 1,z j ) hs length t lest 2. Agin, Clim 1, the reslt follows. This onldes the proof of Item (i). Now we proe item (ii). Sppose tht C hs length t most 5 nd tht P ontins t most 3 hords of C. If C hs length 3, then P ontins no hord of C nd the reslt follows item (i). Ths, sppose tht C hs length 4. B Clim 1, we m ssme tht { 1, 3 } V(P), nd hene let z i { 1, 3 }. If P(z i 1,z i ) hs length 1, then P(z i 1,z i ) = 1 3 nd z i 1 { 1, 3 } z i. Ths, if z i = 1 nd z j = 3, then either P(z i 1,z i ) or P(z j 1,z j ) hs length t lest 2, nd the reslt follows item (i). z z 1 z 2 2 z 1 1 z 2 2 z z 0 0 z 3 z 0 0 z 3 0 k 0 k Figre 3: The illstrtion on the left shows the pth P nd the le C of length 4 in red nd lk, respetiel, while the illstrtion on the right shows the pths P 1 nd P 2 in red nd lk, respetiel. Therefore, we m ssme tht C hs length 5. B Clim 1, we n ssme tht { 1, 4 } V(P). Let {z i,z j } = { 1, 4 }, Sppose, withot loss of generlit, tht i < j nd tht z i = 1 nd z j = 4. B Clim 1, P(z i 1,z i ) nd P(z j 1,z j ) he length 1, i.e., P(z i 1,z i ) = z i 1 z i nd P(z j 1,z j ) = z j 1 z j re two different hords of C in P. We diide this proof into two ses, depending on whether z i z j / E(P) or z i z j E(P). First, sppose tht z i z j / E(P). Sine P(z i 1,z i ) nd P(z j 1,z j ) he length 1, z i 1 = 3 nd j 1 = 2. Sine there re no other erte in C, we he i = 2 nd j = 4. Hene, z 1 = 3, z 2 = 1, z 3 = 2, nd z 4 = 4. Sine 1 2 E(C), the spth P(z 2,z 3 ) hs length t lest 2. Let z e the neighor of 1 in P(z 2,z 3 ), nd pt P 1 = z P(z 0,z 1 ) P(z 4, k ), nd P 2 = G E(P 1 ). Agin, {P 1,P 2 } is pth deomposition of G s desired (see Figre 4). z z z 1 z z z z z 1 z 2 1 z 2 1 z 2 z 2 z 0 0 z 0 0 z 4 4 k 0 z 0 0 z 4 4 k z 0 0 z k 0 0 z 3 4 k () () Figre 4: The illstrtion on the left of figres () nd () shows the pth P nd the le C of length 5 in red nd lk, respetiel, while the illstrtion on the right shows the pths P 1 nd P 2 in red nd lk, respetiel. 6

7 Ths, we m ssme tht z i z j E(P). Then j = i + 1 nd, sine P(z i 1,z i ) is hord, z i 1 = 3. Here we he three ses, depending on whether () 2 / V(P); () z r = 2 nd r < i 1; or () z r = 2 nd r > i+1. In se (), we he z 1 = 3, z 2 = 1, nd z 3 = 4. Therefore, P 1 = P( 0, 0 ) nd P 2 = G E(P 1 ) re two pths whih deompose G (see Figre 4). In se (), we he z 1 = 2, z 2 = 3, z 3 = 1, nd z 4 = 4. Sine 2 3 E(C), the spth P(z 1,z 2 ) hs length t lest 2. Let z e the neighor of 3 in P(z 1,z 2 ). Therefore, P 1 = P( 0,z 1 ) z nd P 2 = G E(P 1 ) re two pths whih deompose G (see Figre 5). In se (), we he z 1 = 3, z 2 = 1, z 3 = 4, ndz 4 = 2. Sine P ontins tmostthreehords, tmostoneetween P(z 0,z 1 ) nd P(z 3,z 4 ) is hord. B smmetr, we n sppose tht P(z 3,z 4 ) is not hord, nd let z e the neighor of 2 in P(z 3,z 4 ), Therefore, P 1 = P( 0,z 0 ) z nd P 2 = G E(P 1 ) re two pths whih deompose G (see Figre 5). 2 z 1 z 1 z 3 z 2 3 z 0 z k 2 z 1 z 1 z 3 z 2 3 z 0 z k 0 1 z 2 k 2 z 4 3 z 1 z z z z 2 k 2 z 4 3 z 1 z z z 3 () () Figre 5: The illstrtion on the left of figres () nd () shows the pth P nd the le C of lenght 5 in red nd lk, respetiel, while the illstrtion on the right shows the pths P 1 nd P 2 in red nd lk, respetiel. Note tht K5 n e deomposed into pth P nd le C sh tht P ontins preisel for hords of C. Therefore, Lemm 2.3(ii) is tight. Corollr 2.4. Let G e onneted grph tht n e deomposed into non-empt grph H nd k pirwise erte-disjoint les of length 3 or 4. Then pn(g) pn(h)+k. Proof. The proof follows indtion on k. Let {H,C 1,...,C k } e deomposition s in the sttement. If k = 1, then let D e minimm pth deomposition of H, nd let P e pth of D tht interepts C 1. B Lemm 2.3(ii), G = P +C 1 dmits deomposition into two pths, s P 1,P 2. Therefore, D = D P +P 1 +P 2 is pth deomposition of G withsizepn(h)+1. Nowsppose k > 1. Sine Gisonneted, we hev(c i ) V(H), for eer i {1,...,k}. Ths, H = H +C k is onneted. From the se k = 1 we he pn(h ) pn(h)+1. Now, note tht {H,C 1,...,C k 1 } is deomposition of G s in the sttement. B the indtion hpothesis, pn(g) pn(h )+k 1 pn(h)+k. The net reslt is ersion of Lemm 2.3 for K 5 nd K 5. Lemm 2.5. If G is onneted grph tht n e deomposed into pth nd op of K 5 or op of K5, then pn(g) = 3. Proof. Let P e pth nd H e op of K 5 or K 5, s in the sttement. It is ler tht H dmits deomposition into le C, nd pth or le B. If H is isomorphi 7

8 to K5, then P ontins t most one hord of C. B Lemm 2.3(i), P + C dmits deomposition into two pths, s P 1,P 2. Therefore, {P 1,P 2,B} is pth deomposition of G with size 3. If H is isomorphi to K 5, then P ontins no hord of C or B. B Lemm 2.3(i), P + C dmits deomposition into two pths, s P 1,P 2, sh tht P 1 ontins etl one edge of C. B Lemm 2.3(i), P 1 +B dmits deomposition into two pths, s P 3,P 4. Therefore, {P 2,P 3,P 4 } is pth deomposition of G with size 3. The net reslt is n nlogos ersion of Corollr 2.4 for K 5 nd K5. Its proof ses Lemm 2.5 insted of Lemm 2.3(ii). Corollr 2.6. Let G e onneted grph tht n e deomposed into non-empt grph H nd k pirwise erte-disjoint opies of K 5 or K 5. Then pn(g) pn(h)+2k. The following reslt shows tht reding sgrphs n sor opies of K 3,K 5 nd K 5 while keeping its reding propert. From now on, gien grph G nd sgrph H of G, we denote C H 3 (resp. C H 5 ) the set of omponents of G E(H) isomorphi to K 3 (resp. K 5 or K 5 ). We denote C 3 (resp. C 5) the omponents of G isomorphi to K 3 (resp. K 5 or K 5 ). Lemm 2.7. Let G e grph sh tht C3 = C 5 =, nd let H e r-reding sgrph of G. Then, H = H + C C C is n ( r+ C 3 H CH 3 H +2 CH 5 ) -reding sgrph of G. 5 Proof. Let G, H, nd H e s in the sttement. Let I H nd I H e the isolted erties of G E(H) nd G E(H ), respetiel, nd let r = r + C3 H + 2 CH 5. Sine H is n r-reding sgrph of G, we he pn(h) r nd I H 2r. Sine C3 = C 5 =, eh omponent in C3 H C5 H interset H. B ppling Corollries 2.4 nd 2.6 in eh omponent of H, we he pn(h ) pn(h)+ C3 H +2 CH 5 r. Note tht, if is erte in V ( C C C ), then is not isolted in G E(H), hene / I 3 H CH H, t is isolted 5 in G E(H ). Ths, we he I H = I H +3 C H 3 +5 CH 5 2(r + CH 3 +2 CH 5 ) 2r. Therefore, H is n r -reding sgrph of G. The net two reslts show tht, lthogh pn(k 5 ) = 3, n proper sdiision of K 5 n e deomposed into two pths. Proposition 2.8. If G is grph otined from K 5 sdiision of one of its edges, then pn(g) = 2. Proof. Let G e s in the sttement. Let, nd,,z e the erties of G with degree 3 nd 4, respetiel, nd let w e the erte of degree 2 of G. There re two ses, depending on whether w hs neighor of degree 3. First, sppose tht w hs neighor of degree 3. Sine the erties of degree 3 in K 5 re re not djent, the other neighor of w is erte of degree 4. We m sppose, withot loss of generlit, tht N(w) = {,}. In this se, {wz,wz} is pth deomposition of G with size 2 (see Figre 6). Ths, we m sppose tht the two neighors of w he degree 4. Sppose, withot loss of generlit, tht N(w) = {,z}. In this se, {wz,wz,} is pth deomposition of G with size 2 (see Figre 6). Corollr 2.9. If G is proper sdiision of K 5, then pn(g) = 2. 8

9 w z w z () () Figre 6: Deompositions into two pths of the grph otined from K5 of its edges. sdiiding one 2.1 Liftings In this setion, we present two reslts tht llow s to otin reding sgrphs in sittions where we se liftings. Let,,z e three erties in grph G. We s tht z is lid lifting if,z E(G) nd z / E(G). In this se, the lifting of,z t is the opertion of remoing the edges nd z nd dding the edge z, whih ields the grph G = G z + z. Alterntiel, we denote G G + z nd G G z. If is erte of degree 2, the (onl) lifting t is lled sppression of. Lemm Let G e onneted grph nd P G e pth. Let,, V(G) sh tht {,,} V(P) nd tht ẑ is lid lifting in G E(P). If t lest two erties of G re isolted in G = (G E(P))+ nd eer omponent of G is Glli grph or isomorphi to K 3 or to K 5, then G ontins reding sgrph. Proof. Let G,P,G nd,, e s in the sttement. Let C e the omponent of G ontining, nd let C = C. Pt H = P + C nd r = V(C ) / If C is Glli grph, then pn(c) pn(c ) V(C ) /2. If C K 5, then C is sdiision of K 5 nd, Corollr 2.9, pn(c) = 2 = V(C ) /2. In these ses, we he pn(h) pn(c) + pn(p) V(C ) /2 +1 = r. If C K 3, then C is le of length 4. Sine {,,} V(P), the grph H is onneted nd, Corollr 2.4, pn(h) = 2 = 1 + V(C ) /2. Now, note tht G E(P) E(C) = G E(C ) = G. Sine G hs t lest two isolted erties nd C is omponent of G, the grph G hs t lest 2+ V(C ) 2r isolted erties. Hene, H is n r-reding sgrph of G. Lemm Let G e onneted grph nd P G e pth. Let,,,,, V(G) e sh tht, {,,} V(P) nd {,,} V(P), nd tht,â re lid liftings in G E(P). Moreoer, sppose tht {,} N(), {,} N() 1. If t lest two erties of G re isolted in G = (G E(P)) + + â, nd eer omponent of G is Glli grph or isomorphi to K 3 or to K5, then G ontins reding sgrph. Proof. Let G,P,G nd,,,,, e s in the sttement. First, sppose tht there is onl one omponent, s C, of G ontining the edges nd. Let C = C â nd H = P + C. Sine {,,} V(P) nd {,,} V(P), the grph H is onneted. Pt r = V(C ) /2 +1. If C is Glli grph or C K 5, then pn(c) pn(c ) V(C ) /2. Ths, pn(h) pn(c)+pn(p) V(C ) /2 +1 = r. If C K 3, thenc isleoflength5nd,,,,, V(C)(inthisse, wehe{,} {,} 9

10 ) nd P hs t most three hords of C, ese {,} N(), {,} N() 1. Ths, Lemm 2.3(ii), pn(h) = 2 = 1+ V(C ) /2 = r. In eh of the ses oe we he pn(h) r. Now, note tht G E(P) E(C) = G E(C ) = G. Sine G hs two isolted erties nd C is omponent of G, the grph G hs 2+ V(C ) 2r isolted erties. Therefore, H is n r-reding sgrph of G. Now, sppose tht C 1 nd C 2 re two distint omponents of G ontining, respetiel, nd. Let C 1 = C 1, C 2 = C 2 â nd let H = P + C 1 + C 2. Pt r = V(C 1) /2 + V(C 2) /2 +1. Welimtht H isnr-reding sgrphofg. Anlogoslto these oe, we he G E(P) E(C 1 ) E(C 2 ) = G E(C 1 ) E(C 2 ) = G, nd G hs t lest 2 + V(C 1 ) + V(C 2 ) 2r isolted erties. Also, if C i K5 or C i is Glli grph, we he pn(c i ) V(C i) /2, for i = 1,2. If C i K 3, then C i is le of length 4. Pt H = P + C 1. Sine,, V(C 1 ) nd {,,} V(P), H is onneted. Agin, we he pn(h ) 1 + V(C 1 ) /2. (if C 1 K 3, we se Lemm 2.3(ii)). Now, note tht H = H + C 2 is onneted, ese,, V(C 2 ) nd {,,} V(P). Anlogosl, if C 2 K 5 or C 2 is Glli grph, pn(h) pn(h ) + pn(c 2 ) 1 + V(C 1) /2 + V(C 2) /2 r; nd if C 2 K 3, then Lemm 2.3(ii), we he pn(h) pn(h ) V(C 1 ) /2 + V(C 2 ) /2 r. Therefore, H is n r-reding sgrph of G. 3 Grphs with treewidth t most three In this setion we erif Conjetres 1 nd 2 for grphs with treewidth t most 3. Let k e positie integer. It is known tht grphs with treewidth t most k re preisel the prtil k-trees [2]. A grph G is k-tree if one of the following onditions holds: (i) G is isomorphi to K k, or (ii) G ontins erte sh tht G[N()] K k nd G is k-tree. A erte of k-tree G is terminl erte or, simpl, terminl, of G if d() = k or G K k. It is not hrd to hek tht if G is k-tree with t lest k+1 erties nd is terminl of G, then G is k-tree. Therefore, it is possile to otin op of K k from n k-tree seqene of remols of terminl erties. The following fts show tht eer k-tree with t lest k + 2 erties ontins t lest two non-djent terminls. Ft 1. If G is k-tree with t lest k +2 erties, then eer pir of terminls of G is non-djent. Proof. Let G e k-tree with t lest n k + 2 erties nd let e terminl of G. Sine is terminl, G = G is k-tree. Let e neighor of. Sine G hs t lest k+1 erties, d G () k, whih implies d G () k+1, hene is not terminl. Ft 2. If G is k-tree with t lest k+2 erties, then G ontins t lest two terminls. Proof. Let G e k-tree with t lest n k+2 erties. The proof follows indtion on n. If n = k + 2, then G is isomorphi to K k+2 e, nd the sttement holds. Ths, sppose n > k+2 nd let e terminl of G. B the definition of terminl, G = G is k-tree. Sine eer pir of neighors of is djent, Ft 1, is djent to t most one terminl of G. B the indtion hpothesis, there is t lest one terminl, s, of G whih is not djent to. Therefore, is terminl of G. 10

11 We s tht grph G is prtil k-tree if it is sgrph of k-tree G (see [1, 2]). In this se, we s tht G is n nderling k-tree of G. The net ft shows tht there is n nderling k-tree G of G sh tht G ontins eer terminl erte of G. In ft, one n proe tht if G hs t lest k erties, then G is spnning sgrph of k-tree, t, in this pper, we do not mke se of this ft. Ft 3. If G is prtil k-tree with t lest k erties, then there eists n nderling k-tree G of G sh tht G ontins eer terminl erte of G. Proof. Let G e s in the sttement nd let G e n nderling k-tree of G with minimm nmer of erties. Sppose, for ontrdition, tht there is terminl of G sh tht / V(G). Then G is n nderling k-tree of G, ontrdition to the minimlit of G. Prtil 3-trees re lso known their foridden minors hrteriztion [1]. Therefore, eer sgrph of prtil 3-tree is lso prtil 3-tree, nd the grph otined from prtil 3-tree G sppressing erte of degree 2 is lso prtil 3-tree. Let G e prtil k-tree, nd let G e n nderling k-tree of G. We s tht erte of G is terminl if is terminl erte in G. Therefore, if G is prtil k-tree with t lest k + 2 erties, then G ontins t lest two non-djent terminls. The following ft ot prtil k-trees will e sed often in the proof of Theorem 3.2. Ft 4. Let G e prtil k-tree with t lest k+1 erties, e terminl of G, nd S e set of edges joining erties in N() sh tht S E(G) =. Then G +S is prtil k-tree. Proof. Let G, k, nd S e s in the sttement, nd let G e the nderling k-tree of G. Notetht G isnnderling k-treeof G. Sine N G () K k, we he S E(G ). Ths, G is n nderling k-tree of G +S. 3.1 Dole entered 3-trees In this setion, we present nd hrterize fmil of 3-trees tht will e sefl in the proof of Theorem 3.2. We s tht 3-tree G is dole entered if there re two erties, V(G) sh tht eer terminl erte of G different from, is djent to nd (see Figre 7). In this se, we s tht, re the enters of G. The hrteriztion of dole-entered 3-trees presented here is gien in terms of grph joins. Gien two ertedisjoint grphs H 1 nd H 2, we s tht grph G is the join of H 1 nd H 2, denoted H 1 H 2, if G is the grph otined from H 1 +H 2 joining eer erte of H 1 to eer erte of H 2, i.e., V(G) = V(H 1 ) V(H 2 ) nd E(G) = E(H 1 ) E(H 2 ) {: V(H 1 ), V(H 2 )}. Proposition 3.1. Let G e dole entered 3-tree with enters nd. There is tree T sh tht G T K 2, where V(T) = V(G). Moreoer, the terminls of G re preisel the lefs of T. Proof. Let G nd, e s in the sttement. The proof follows indtion on the nmer n of erties of G. If n = 3, then G K 3, nd the reslt follows with T K 1. 11

12 Now, sppose n 4, nd let e terminl erte of G different from,. B the definition of dole entered 3-trees, we he, N(). Let e the neighor of different from,, nd let G = G. Sine G is 3-tree, we he N() K 3, hene, N(). Note tht tht eer terminl of G, eept for, is terminl of G. The onl possile terminl of G (different from,) tht is not terminl of G is. Sine N() indes omplete sgrph of G, is djent to,. Ths eer terminl of G different from, is djent to nd, hene G is dole entered 3-tree with n 1 erties. B the indtion hpothesis, there is tree T sh tht G T K 2. Let T e the the tree otined from T dding the erte nd the edge. Clerl, we he G = T K 2. Figre 7: emples of dole entered 3-trees T K 2 where T is illstrted in red. 3.2 Glli s Conjetre for Grphs with Treewidth t most 3 In this setion, we erif Conjetre 1 for grphs with treewidth t most 3. In ft, we proe slightl stronger reslt. We proe tht if G is grph with n erties nd treewidth t most 3, then pn(g) n/2, or G is isomorphi to K 3 or to K5, whih is the grph otined from K 5 the remol of one edge. The proof onsists of showing tht minimm onter-emple for this sttement is either n odd grph or sgrph of dole entered 3-tree in whih its enters he odd degree. This implies ontrditions to Theorem 1.1 nd to Theorem 1.2, respetiel. Let G e grph, let e E(G) e t-edge, nd let C 1 nd C 2 e the omponents of G e. We s tht e is sefl if C 1 nd C 2 he t lest two erties, otherwise we s tht e is seless. Now, we re le to proe or min reslt of this setion. Theorem 3.2. Let G e onneted prtil 3-tree with n erties. Then pn(g) n/2 or G is isomorphi to K 3 or to K 5. Proof. Let G nd n e s in the sttement. Sppose, for ontrdition, tht the sttement does not hold, nd let G e onter-emple for the sttement tht minimizes n. It is not hrd to hek tht n 6. Clim 1. G ontins no reding sgrph. Proof. Sppose, for ontrdition, tht G ontins n r-reding sgrph H. Sine G is onneted nd n 6, we he C3 = C5 =. Ths, Lemm 2.7, H = H + C C C 3 H CH 5 is n ( r+ C3 H +2 CH 5 ) -reding sgrph of G. Moreoer, no omponent of G E(H ) is isomorphi to K 3 or to K5, ths H is Glli grph. B Lemm 2.1, G is Glli grph. 12

13 Clim 2. G ontins no sefl t-edge. Proof. Sppose, for ontrdition, tht e = 1 2 is sefl t-edge in G. For i = 1,2, let G i e the omponent of G e tht ontins i, nd let G i = G i+e. Note tht G i is sgrph of G, hene G i is prtil k-tree. Let n i = V(G i ). Sine e is sefl t-edge, n i < n, nd sine G i hs erte with degree one, G i is not isomorphi to K 3 or to K5. Ths, the minimlit ofg, thegrphg i is Glligrph, fori = 1,2. Let D i epth deomposition of G i sh tht D i n i /2, nd let P i e the pth in D i tht ontins the edge e. Pt P = P 1 +P 2, nd D = D 1 P 1 +D 2 P 2 +P. Note tht the erties 1 nd 2 re the onl erties of G in oth G 1 nd G 2, hene n 1 +n 2 = n+2. Therefore, we he D = D 1 + D 2 1 n 1 /2 + n 2 /2 1 (n 1 +n 2 2)/2 = n/2. Clim 3. If is erte of degree 2 in G nd N() = {,}, then E(G). Proof. Sppose for ontrdition tht / E(G), nd note tht G = G + is prtil 3-tree. Sine n 6 either G is Glli grph or G K 5. If G is Glli grph, then pn(g) pn(g ) (n 1)/2. If G K 5, then G is sdiision of K 5 nd, Corollr 2.9, pn(g) = 2 = (n 1)/2. In eh se we otined ontrdition to the minimlit of G. Clim 4. If {,d} E(G) is n edge-t, then {,,,d} indes le in G. Proof. LetG = G d. IfG ontinsmorethntwoomponents, thenothndd re t-edges, hene {, d} is not n edge-t ( miniml edge seprtor). Ths, let G 1,G 2 ethetwo omponentsofg. Sppose, withotlossofgenerlit, tht, V(G 1 ) nd,d V(G 2 ). Ths, we he,d,,d / E(G). If {,,,d} does not inde le in G, then we n sppose tht d nd d / E(G). Let G 2 = G 2 +d. B the minimlit of G, G 2 is Glli grph or is isomorphi to K 3 or to K5. If G 2 K 3, then let e the erte of G 2 different from nd d. Note tht in G, the erte hs degree two nd hs non-djent neighors, ontrdition to Clim 3. Ths, G 2 is Glli grph or G 2 K5. Let G 1 = G d, nd let P e pth in G 1 joining to d. Let H = G 2 + P, nd note tht H is proper sdiision of G 2 (see Figre 8). Let r = V(G 2 ) /2. Note tht G E(H) hs t lest V(G 2 ) 2r isolted erties. If G 2 is Glli grph, then pn(h) pn(g 2 ) V(G 2 ) /2 = r. If G 2 K5, then Proposition 2.8, we he pn(h) 2 = V(G 2) /2 = r. Ths, H is n r-reding sgrph, ontrdition to Clim 1. Let G e n nderling 3-tree of G. Sine G G nd n 6, G hs t lest 6 erties. B Fts 1 nd 2, G hs t lest two non-djent terminls, s nd. Rell tht d G () = d G () = 3. Ths, d G (),d G () 3. The net lim shows tht these terminls mst he degree preisel 3. Clim 5. Eer terminl of G hs degree 3. Proof. Let nd e two (non-djent) terminls of G, where d() d(). Sppose for ontrdition tht d() 2. If d() = d() = 1, then n pth joining nd is reding pth, ontrdition to Clim 1. Sppose tht d() = 2, nd let N() = {,}. Sppose lso tht nd he t most one neighor in ommon, nd let P e shortest pth joining nd. Sppose, withot loss o generlit, tht is the neighor of in P, 13

14 G 1 G 2 d Figre 8: the two omponents G 1 nd G 2 of G re illstrted the regions, while the pth P is olored red. nd let P e the grph otined from P the ddition of the edges inident to or tht re not in P (see Figre 9). Sine P is shortest pth nd nd he t most one neighor in ommon, P is pth, nd sine nd re isolted in G E(P), P is reding pth, ontrdition to Clim 1. Ths, we m sppose tht nd he preisel two neighors, s nd, in ommon. Pt G = G. BClim 3, we he E(G), hene G is onneted nd V(G ) 4. Ths, G K 3. B Ft 4, G is prtil 3-tree. If G K 5, then G = G â = G ++ is proper sdiision of K 5, nd Corollr 2.9 we he pn(g ) = 2. Let T e the tringle with the edges,,, nd note tht G = G + T. B Corollr 2.4, pn(g) 3 = 7/2 = n/2. Ths, we m sppose tht G is Glli grph nd, therefore, pn(g ) (n 2)/2. B Corollr 2.4, pn(g) pn(g )+1 n/2. () () Figre 9: the pth P joining nd, tht n e etended to pth dding the possile remining edge. Ths, we n sppose tht d() = 3, nd let N() = {,,}. First, we lim tht tht G is onneted. Indeed if G is not onneted, then is inident to t-edge, s. B Clim 2, is seless t-edge, hene d() = 1. Sine nd re not djent, we he. Agin, if P is minimm pth joining to, nd P is the pth otined from P dding the possile remining edge inident to (see Figre 9), then nd re isolted in G E(P), hene P is reding pth, ontrdition to Clim 1. Now, we lim tht N() indes liqe in G. Indeed, if N() does not indes liqe in G, then there eists pir of non-djent erties in N(), s nd. Let P e minimm pth in G joining nd. Sine P is minimm pth, it ontins onl one neighor of, s. If d() = 2, then let e the neighor of different from, nd pt P = P ++. If d() = 1 pt P = P + (see Figre 10). Let G = (G E(P))+. B Ft 4, G is prtil 3-tree, nd the minimlit of G, 14

15 eer omponent of G is Glli grph, or is isomorphi to K 3 or to K 5. Moreoer, nd re isolted in G. Therefore, Lemm 2.10, G ontins reding sgrph, ontrdition to Clim 1. Therefore, we m ssme tht N() indes liqe in G. () () () Figre 10: Figres ()-() show the pth P, in red, whih its remol llows s to ppl lifting nd then Lemm 2.10, to otin reding sgrph of G. LetP eminimmpthfromn()to ing. Sppose, withotlossofgenerlit, tht P is pth from to. Sine P is minimm, it does not ontin nd. Let e the neighor of in P. Sppose tht N() N() 1. If d() = 2, then let e the neighor of different from, nd pt P = P If d() = 1, then pt P = P (see Figre 10). Pt G = (G E(P))+â. Anlogosl to the se oe, G ontins reding sgrph, ontrdition to Clim 1. Ths, we m sppose tht N() N() = 2. Sppose tht, N(). Let P = (see Figre 10) nd let G = (G E(P))+â. Anlogosl to the ses oe, G ontins reding sgrph, ontrdition to Clim 1. In wht follows, we present three properties of G. Let nd e two non-djent terminl erties. Propert 1 sttes tht nd he t lest two neighors in ommon, nd Propert 2 sttes tht nd do not he (ll) three neighors in ommon. Finll, Propert 3 sttes tht eer ommon neighor of two terminl hs odd degree, whih enles s to hrterize the minimm onter-emple, nd show tht it stisfies the sttement, otining ontrdition. Propert 1. There re no two terminls with t most one neighor in ommon. Proof. Sppose, for ontrdition, tht there re two terminls, s nd, sh tht N() N() 1, nd let G = G, G = G nd G = G. Clim 6. The grph G is disonneted. Proof. Sppose for ontrdition tht G is onneted. First, sppose tht N() nd N() re not liqes in G. Ths, there re erties, N() nd, N() sh tht, / E(G). Let nd z e the remining erties in N() nd N(), respetiel. Sine G is onneted, there is pth P from to z in G. Let P = P ++z (see Figre 11), nd pt G = (G E(P))+â+. It is ler tht nd re isolted in G. Moreoer, we he {,} N(), {,} N() 1, ese N() N() 1. Therefore, Lemm 2.11, G ontins reding sgrph, ontrdition to Clim 1. Ths, we n ssme with withot loss of generlit tht N() is liqe. Now, sppose tht N() is not liqe in G, nd let nd e two nondjent erties in N(). Let e the remining erte in N(), nd let P e minimm pth in G from to erte, s, in N(). Let nd z e the remining erties of N(), nd 15

16 z z z z () () () (d) Figre 11: Figres ()-(d) show the pth P, in red, whih its remol llows s to ppl lifting nd then Lemm 2.11, to otin reding sgrph of G. let P = + P + + z + z (see Figre 11). Clerl, P is pth in G. Now let G = (G E(P)) + â +. Anlogosl to the se oe, G ontins reding sgrph, ontrdition to Clim 1. Finll, we n ssme tht N() is liqe, nd let P e minimm pth from erte in N() to erte in N(). We m sppose, withot loss of generlit, tht P joins nd. Pt P = P + + z + z (see Figre 11) nd G = (G E(P)) + â +. Anlogosl to the ses oe, G ontins reding sgrph, ontrdition to Clim 1. From now on, we fi minimm pth P in G joining erte in N() to erte in N(). We m sppose, withot loss of generlit, tht P joins nd, nd pt P = +P +. Let nd e the remining erties of N(), nd let nd z e the remining erties of N(). We lim tht t lest one edge etween nd z elong to E(G). Indeed, sppose tht the edges,z / E(G) (see Figre 11d). Ths, let G = (G E(P))+ +ŷz. Sine N() N() 1, we he {, } N(), {, } N() 1. Anlogosl to the proof of Clim 6, G ontins reding sgrph, ontrdition to Clim 1. Therefore, G ontins t lest one of the edges or z. Assme, withot loss of generlit, tht G ontins the edge z. Clim 7. nd re not t-erties. Proof. Sppose for ontrdition tht is t-erte. Sine d() = 3, t lest one of the edges inident to mst e t-edge. Sine z E(G), the edges,z re not t-edges, hene is t-edge. Sine P + joins to, the omponent of G tht ontins is not triil. Therefore, is sefl t-edge, ontrdition to Clim 2. To proe tht is not t erte, we first show tht nd he degree t lest 2. If d() = 1, then, / E(G). Sine is not t-erte, there is minimm pth Q in G joining erte in {,z} to erte in {,}. We m ssme, withot loss of generlit, tht Q is pth joining nd. B the minimlit of Q we he / V(Q ). Sppose for ontrdition tht / E(G). Let Q = Q + + z + z (see Figre 12), nd let G = (G E(Q)) + â +. Sine N() N() 1, we he {,} N() 1 nd {,} N() 1. B Lemm 2.11, G ontins reding sgrph, ontrdition to Clim 1. Ths, we n sppose tht E(G). In this se, let Q = P +++z+z (see Figre 12), nd let G = (G E(Q))+ +. Agin, Lemm 2.11, G ontins reding sgrph, ontrdition to Clim 1. B smmetr we he d() > 1. Now, sppose tht is t-erte. Sine the erte d() = 3, t lest one of the edges inident to is t-edge. Sine is djent to nd is n end-erte of P, we 16

17 z z () () Figre 12: Figres () nd () show the pth P, in red, whih its remol llows s to ppl lifting nd then Lemm 2.11, to otin reding sgrph of G. he d() > 1. Ths, sine d() 2 nd d() 2, if e is t edge inident to, then e is sefl, ontrdition to Clim 2. Sine P is pth joining nd in G, the erties nd re in the sme omponent, s C 1, of G. Sine is not t-erte of G, there is omponent, s C 2 C 1, of G ontining nd z. Note tht C 1 nd C 2 re the onl omponents of G, sine G is onneted, G disonneted, nd nd z re djent in G. Sine is not t-erte ofg, mst he neighor inc 2. Sppose, withot loss of generlit, tht V(C 2 ). Now, sppose tht V(C 2 ), hene {,} is n edge-t of G (see Figre 13). Sine / E(G), {,,,} does not inde le in G, ontrdition to Clim 4. Ths, we n sppose tht V(C 1 ). Hene, {,} is n edge-t (see Figre 13). B hpothesis, the onl possile neighor in ommon etween nd is. Ths, / N(), whih mens tht / E(G). Therefore, {,,,} does not inde le in G, ontrdition to Clim 4. C 1 C 1 C 2 z C 2 z () () Figre 13: the possile omponents of G. Propert 2. There re no two terminls with (ll) three neighors in ommon. Proof. Sppose for ontrdition tht nd re two non-djent terminls sh tht N() = N() = {,,}. Clim 8. {,,} indes liqe. Proof. Sppose forontrditiontht{,,}doesnot indeliqe. Wensppose, withot loss of generlit, tht / E(G). We lim tht d() is odd. Indeed, sppose d() is een, nd let G = G +. B Ft 4, G is prtil 3-tree. It is ler tht d G () is odd. Ths, the omponent C of G tht ontins is not isomorphi to K 3. Ths, the minimlit of G, C is Glli grph or C K5. Now, pt C = C â, nd note tht pn(c) V(C ) /2 (if C K5, then we se Proposition 2.8). Let D e minimm pth deomposition of C. Sine d G () = d C () = d C () is odd, t lest one 17

18 pth P D hs s n end-erte. Note tht Q = P + nd R = re pths of G (see Figre 14). Therefore, we he pn(c+) = pn(c). Let H = C++R, we he pn(h) pn(c + )+1 V(C ) / Finll, note tht, nd eer erte in V(C ) is isolted in G E(H), hene H is reding sgrph of G, ontrdition to Clim 1. Therefore, d() is odd. B smmetr, d() is odd. G G () () Figre 14: the pths Q nd R re illstrted, respetiel, in lk nd red. Now, let G = G. It is not hrd to hek tht if G is disonneted, then for some {,,} the set {,} is 2-edge-t, t sine / E(G), {,,} does not inde le in G. Sine G is sgrph of G, G is prtil 3-tree, nd the minimlit ofg,g isglligrphorisomorphi tok 3 ortok 5. Sinendhe odd degrees in G, G is not isomorphi to K 3. First, sppose tht G is Glli grph, nd let D e minimm pth deomposition of G. Let P,P D e pths ontining nd s end-erties, respetiel. Pt P = P +, P = P +, nd R = (if P = P, we pt P = P = P ++) (see Figre 14). Clerl, D = D P P +P +P +R is pth deomposition of G sh tht D V(G ) /2 +1 n/2. Therefore, we m ssme tht G K 5. nd let V(g ) = {,,,,}. Let R 1 =,R 2 =,R 3 =, nd note tht {R 1,R 2,R 3} is pth deomposition of G. Notetht R i does notontin or, fori = 1,2,3, R 1 doesnot ontin, ndr 2 doesnot ontin. Pt R 1 = R nd R 2 = R Note tht {R 1,R 2,R 3 } is deomposition of G (see Figre 15). Therefore, pn(g) = 3 = 7/2 = n/2. This finishes the proof of the lim. () () Figre 15: in Figre (), the pths R 1, R 2, nd R 3 re illstrted, respetiel, in red, lk, nd le; in Figre (), the pth P nd the lifted edges re illstrted, respetiel, in red nd le. Now, let P = (see Figre 15) nd G = (G E(P))+ +â. B Ft 4, G is prtil 3-tree. Moreoer, G is onneted nd, minimlit of G, G is Glli grph 18

19 or isomorphi to K 3 or to K 5. Sine n 6, we he V(G ) 4 nd therefore G K 3. If G K 5, then n = 7 nd G â = G E(P) is proper sdiision of K 5. B Proposition 2.8, pn(g E(P)) = 2. Hene, pn(g) = 3 = n/2. If G is Glli grph, then pn(g ) V(G ) /2 nd pn(g) V(G ) /2 +1 = n/2. This onldes the proof of Propert 2. Propert 3. If is ommon neighor of two terminls, then d() is odd. Proof. From the Properties 1 nd 2, if, re two terminls of G, we he N() N() = {,}. Here, we proe tht d(),d() re odd. Let nd e two terminls of G, let N() = {,,},N() = {,,d}, nd O = +++++d. Sppose for ontrdition tht d() is een. Now, sppose tht / E(G). Let G = G +. It is ler tht d G () = d G () 1, hene d G () is odd. Let C e the omponent of G ontining the edge. Note tht C K 3 ese d C () = d G () is odd. Now, pt C = C â, nd note tht pn(c) V(C ) /2, (if C K 5, then we se Proposition 2.8). Note tht d C() = d C (), ths, if D C is minimm pth deomposition of C, there is pth P in D C tht hs s n end-erte. Note lso tht C does not ontin. Therefore, P = P + is pth. Pt Q = d (see Figre 16) nd note tht D C P +P +Q is pth deomposition of H = C +O. Moreoer, the erties nd, nd the erties in V(C ) re isolted in G E(H), nd pn(h) pn(c) + 1 V(C ) / Therefore, H is ( V(C ) /2 + 1 ) -reding sgrph of G, ontrdition to Clim 1. Therefore, we n sppose tht E(G). B smmetr, we he d E(G). C C d () d () Figre 16: the pths P nd Q re illstrted, respetiel, in lk nd red. Now, let O = d + + d, nd let G = G d. It is ler tht d G () = d G () 3, hene d G () is odd. Let C e the omponent of G ontining the edge. In wht follows, the proof is similr to the se oe. We he C K 3 ese d C () = d G () is odd. Let C = C â nd note tht d C () is lso odd, nd pn(c) V(C ) /2. Let D C e minimm pth deomposition of C, nd let P D C e pth tht hs s n end-erte. Agin, P does not ontin ese / V(C), hene P = P + is pth. Pt Q = d (see Figre 16) nd note tht D C P + P + Q is pth deomposition of H = (C ) + O. Anlogosl to the se oe, H is ( V(C ) /2 +1 ) -reding sgrph of G, ontrdition to Clim 1. Therefore, we onlde tht d() is odd. Now, we n onlde or proof. Let 1,..., k e the terminls of G. Sine n 6, we he k 2. B Clim 5, Propert 1, nd Propert 2, we he N( i ) N( j ) = 2 19

20 for eer i j. In wht follows, the proof is diided into two ses, depending on whether there re two erties, s,, sh tht, N( i ) for eer i = 1,...,k. First, sppose tht sh pir of erties does not eist. Let N( 1 ) = { 1, 2, 3 }. Sine N( 1 ) N( 2 ) = 2, wemsppose, withotlossofgenerlit, thtn( 2 ) = { 1, 2, 4 }, where 4 3. Sine there re no two erties ontined in N( i ), for eer i = 1,...,k, there is terminl, s 3, sh tht 1 or 2 is not ontined in N( 3 ). Sppose, withot loss ofgenerlit, tht 2 / N( 3 ). Ths, N( 3 ) = { 1, 3, 4 }, ese N( 3 ) N( 1 ) = N( 3 ) N( 2 ) = 2. Now, sppose tht there is erte 5 different from 1,..., 4 ontined in the neighor of terminl erte, s. It is not hrd to hek tht N() N( i ) < 2 for t lest one i {1,2,3}, ontrdition to Propert 1. Now we se n nderling 3-tree G of G. Sine 1,..., k re terminls, the set N( i ) indes omplete grph in G, i.e., i j E(G ) for eer i,j {1,2,3,4}, with i j. B Ft 4, G + = G 1 k is 3-tree. If G + hs erte different from 1, 2, 3, 4, then G + ontins t lest 5 erties, nd Fts 1 nd 2, the grph G + ontins two non-djent terminls. Ths, G + ontins terminl + whih is not in { 1, 2, 3, 4 }, t then + is terminl of G, nd, onseqentl, + is terminl of G different from 1,..., k, ontrdition. Therefore, G onsists onl of the erties 1,..., k, 1, 2, 3, 4. B Clim 5, 1,..., k he degree 3. Note tht 1, 2 N( 1 ) N( 2 ), 3 N( 1 ) N( 3 ), nd 4 N( 2 ) N( 3 ). Ths, j is ommon neighor of t lest two terminls of G, for j = 1,...,4. B Propert 3, 1, 2, 3, 4 he odd degree. Therefore, G is n odd Grph, nd Theorem 1.1, we he pn(g) = n/2. We note tht, sing Propert 2, one n proe tht G is grph otined from K 4,4 remoing perfet mthing. Now, sppose tht there re two erties, s,, sh tht, N( i ), for eer i = 1,...,k. Note tht this is lso tre for the nderling spnning 3-tree G of G, i.e., G is dole entered 3-tree with enter in,. Ths, Proposition 3.1, there is tree T sh tht G = T K 2, where V(G ) = V(T)++. Now, if C is le in G, then C mst ontin either or, otherwise V(C) V(T), t T hs no les. It is ler tht if C is le in G, then C is le in G, hene C ontins or. Ths, eer le of G mst ontin or, whih he odd degree Propert 3. Therefore, Theorem 1.2, we he pn(g) = n/2. This onldes the proof. The net orollr omes from the ft tht eer grph with no sdiision of K 4 is prtil 3-tree. In ft, it is not hrd to hek tht grphs with no sdiision of K 4 do not ontin n of the foridden minors for prtil 3-trees. Corollr 3.3. Let G e onneted grph with n erties nd with no sdiision of K 4. Then pn(g) n/2 or G is isomorphi to K Hjós Conjetre for grphs with treewidth t most 3 Before erifing Conjetre 2 for grphs with treewidth t most 3, we gie n nlogos definition of reding sgrph for deling with Conjetre 2. Let G e n Elerin grph, nd let H e n Elerin sgrph of G. Gien positie integer r, we s tht H is n r-le reding sgrph of G if G E(H) hs t lest 2r isolted erties nd n(h) r. If H is n 1-le reding sgrph, we s tht H is reding le of G, nd we s tht H is le reding sgrph if H is n r-le reding sgrph for 20

21 some positie integer r. We s grph G with n non-isolted erties is Hjós grph if n(g) (n 1)/2. The following lemm holds, nd its proof is nlogos to the proof of Lemm 2.1. Lemm 3.4. Let G e n Elerin grph nd H G e le reding sgrph of G sh tht H G. If G E(H) is Hjós grph, then G is Hjós grph. Note tht the sttement of Lemm 3.4 does not reqire G to e onneted. In ft, it is es to hek tht if grph G is erte-disjoint nion of two Hjós grphs, then G is lso Hjós grph. Theorem 3.5. Let G e onneted Elerin prtil 3-trees with n erties. Then n(g) (n 1)/2. Proof. Sppose, for ontrdition, tht the sttement does not hold, nd let G e onter-emple tht minimizes n. The sttement holds triill if n = 3 or n = 4. Ths, we m ssme tht n 5. We lim tht G is 2-onneted. Indeed, sppose tht G ontins t-erte, nd let B 1,...,B k e the 2-onneted omponents of G. It is es to see tht V(G) = ( k i=1 V(B i) ) k+1, i.e., V(G) 1 = ( k i=1 V(B i) ) k. Sine G is miniml, n(b i ) ( V(B i ) 1)/2, for i = 1,...,k. Let D i e minimm le deomposition of B i. Ths, D = k i=1d i is le deomposition of G of size t most k k V(Bi ) 1 2 i=1 i=1 (( k ) ) V(B i ) 1 1 = V(B i ) k = 2 2 i=1 V(G) 1 Therefore, we m ssme tht G is 2-onneted. Sine n 5, G ontins t lest two terminls, s nd, nd sine G is Elerin, d() = d() = 2. Sine G is 2- onneted, there is le C in G ontining nd. The erties nd re isolted in G = G E(C), hene either G = C or C is reding le. Sine G is miniml onter emple, G is Hjós grph, hene Lemm 3.4, G is Hjós grph. Corollr 3.6. Let G e onneted Elerin grph with n erties nd with no sdiision of K 4. Then n(g) n 1/ Grphs with mimm degree 4 In this setion, we erif Conjetres 1 nd 2 for grphs with mimm degree t most 4. Althogh this se ws lred erified [3, 14] for oth onjetres, the proofs presented here eemplifies the pplition of reding sgrphs. 4.1 Glli s Conjetre for grphs with mimm degree 4 Theorem 4.1. Let G e onneted grph with n erties. If G hs mimm degree 4, then pn(g) n/2 or G is isomorphi to K 3, K 5 or to K 5. Proof. Sppose, for ontrdition, tht the sttement does not hold, nd let G e onter-emple tht minimizes n. It is es to hek tht the sttement is tre for 21

22 grphs with t most fie erties. Ths, we m sppose n 6. The following lims follow nlogosl to the the proof of Theorem 3.2, therefore we omit their proofs. Clim 1. G ontins no reding sgrph. Clim 2. G ontins no sefl t-edge. B Corollr 3.3, we m sppose tht G ontins sdiision of K 4. Ths, let H e sdiision of K 4 in G with minimm nmer of edges, nd let 1, 2, 3, 4 e the erties of degree 3 in H. From now on, for eh i,j {1,2,3,4} with i j, let P i,j e the pth in H joining i to j, where for {i,j} {i,j }, the pths P i,j nd P i,j he no internl erte in ommon. Let S e the set of edges in G E(H) inident to 1, 2, 3, 4. Sine G hs mimm degree 4, there is t most one edge in S inident to i, for i = 1,2,3,4. For eh i {1,2,3,4}, if there is n edge e i in S inident to i, then let e i = i z i, nd pt Z = {z i : i z i S}. Clim 3. Z V(H) 1, nd if Z V(H) = {z i }, then z i z j for eer j {1,2,3,4} with i j. Proof. Sppose tht z 1 V(H). If z 1 V(P 1,j ) for n j {2,3,4}, then H + 1 z 1 P 1,j ( 1,z 1 ) is sdiision of K 4 with less edges thn H. Ths, we n sppose, withot loss of generlit, tht z 1 V(P 2,3 ). If P 1,j hs length t lest 2, then H + 1 z 1 P 1,j is sdiision of K 4 with less edges thn H. Therefore, P 1,j = 1 j, for i = 2,3,4. Anlogosl, if z i V(H), then P i,j = i j for j {1,2,3,4} i. Sine z 1 P 2,3, the length of P 2,3 is t lest 2, hene z 2,z 3 / V(H). Therefore Z V(H) 2. Now, sppose z 4 V(H). We he z 4 / V(P 4,j ) for j = 1,2,3. Sine P 1,j hs length 1 for j = 2,3,4, we he z 4 / V(P 1,j ). Therefore, z 4 V(P 2,3 ), hene H + 1 z z 4 2 ontins sdiision of K 4 with less edges thn H. Therefore z 4 / V(H) nd Z V(H) 1. In wht follows we diide the proof on whether Z V(H) = 0 or Z V(H) = 1. (i) Z V(H) = 0. First, sppose tht d S (z i ) = 4 for some i. Ths, we he z 1 = z 2 = z 3 = z 4. We lim tht P i,j hs length t most 1, for eer i,j {1,2,3,4} with i j. Indeed, sppose, withot loss of generlit, tht P 1,2 hs length t lest 2. We he 4 j=2 E(P 1,j) 4. Ths, H + S 1 ontins sdiision of K 4 in G with t most E(H) 1 edges, ontrdition to the minimlit of H. Therefore G is isomorphi to K 5, nd the reslt follows. Now, sppose d S (z i ) 2 nd, withot loss of generlit, sppose tht z 1 z 4 nd z 2 z 3. Let Q 1,4 = z P 1,2 + P 2,3 + P 3,4 + 4 z 4 nd Q 2,3 = z P 2,4 + P 4,1 + P 1,3 + 3 z 3 (we m ignore the edges i z i if i z i / S, see Figre 17). Clerl, D H+S = {Q 1,4,Q 2,3 } is pth deomposition of H + S, nd the erties 1, 2, 3, 4 re isolted in G = G E(H) S. Ths, H + S is 2-reding sgrph of G, ontrdition to Clim 1. Ths, ssme tht d S (z i ) = 3 for some i. We m sppose, withot loss of generlit, tht z 1 = z 2 = z 3. Note tht H = H+ 1 z z z 3 is sdiision of K5, where z 1 nd 4 re its erties of degree 3. B Corollr 2.9, If H is proper sdiision of K5, then pn(h ) = 2. Ths, there eist two pths P nd Q sh tht H = P +Q. Sine 4 hs degree 3 in H, one etween P nd Q, s P, hs 4 s n end-erte. If 4 z 4 S, then pt P = P + 4 z 4. Ths, {P,Q } is deomposition of H +S into two pths, nd the erties 1, 2, 3, 4 re isolted erties of G = G E(H) S. Ths, H + S is 2-reding sgrph of G, ontrdition to Clim 1. 22