Coering planar graphs ith degree bonded forests D. Gonçales LaBRI, U.M.R. 5800, Uniersité Bordea 1, 351 cors de la Libération 33405 alence Cede, France. Abstract We proe that eer planar graphs has an edge partition into three forests, one haing maimm degree 4. his ansers a conjectre of Balogh et al. (J. Combin. heor B. 94 (2005) 147-158).We also proe that eer planar graphs ith girth g 6 (resp. g 7) has an edge partition into to forests, one haing maimm degree 4 (resp. 2). Ke ords: planar graphs, edge partition, forests, trees 1 Introdction A graph G is coered b sbgraphs G 1,...,G k of G if eer edge of G belongs to one of these sbgraphs. A graph G is (t, D)-coerable if it can be coered b t forests and a graph H of maimm degree (H) D. A graph is F(d 1,...,d k )-coerable if it can be coered b k forests F 1,...,F k sch that (F i ) d i for all 1 i k. If d i = the maimm degree of F i is nbonded. B a reslt of Nash-Williams [8], e kno that planar graphs are (3, 0)-coerable (i.e. F(,, )-coerable) and that planar graphs of girth g 4 are (2, 0)-coerable (i.e. F(, )-coerable). In [6], the athors proed that planar graphs are (2, 8)-coerable. he athors also asked hat cold be the minimal d sch that eer planar graph is (2, d)-coerable. In [2], the athors proed that planar graphs are more than (2, 8)-coerable, the are F(,, 8)-coerable. he also proed that there eist non-(2, 3)-coerable planar graphs and the conjectred that planar graphs are (2, 4)-coerable. Or main reslt is slightl stronger than this conjectre. Email address: goncale@labri.fr (D. Gonçales). Preprint sbmitted to Jornal of Combinatorial heor, Series B
l r e Fig. 1. From to e, l and r. heorem 1 Planar graphs are F(,, 4)-coerable. he case of bonded girth planar graphs has also been stdied. It is proen in [6] that planar graphs ith girth at least 5 (resp. 7) are (1, 4)-coerable (resp.(1, 2)-coerable). In [1], the athors proed that planar graphs ith girth at least 10 are (1, 1)-coerable (i.e. F(, 1)-coerable). Here e hae some reslts on forest coerings of planar graphs of girth at least 6 or 7. heorem 2 Planar graphs of girth g 6 are F(, 4)-coerable. heorem 3 Planar graphs of girth g 7 are F(, 2)-coerable. 2 Planar graphs A trianglation is a planar graph in hich eer face is trianglar. In [4] the athor proed that planar graphs are coerable b for forests of caterpillars. His proof orks b indction sing a decomposition of trianglations into three smaller trianglations. We proe heorem 1 sing the same decomposition tool. Consider an embedded trianglation ith at least for ertices and sch that going conter-clockise on the eternal face e sccessiel meet the ertices, and. For a cople (, ) of these ertices e define its partner cople (, ) of ertices. In a trianglation ith at least for ertices, an edge ab is sch that its ends, a and b, hae at least to common neighbors. We consider the seqence of s neighbors going in the clockise sense from to. Let be the second of these ertices being a neighbor of (the first one being, ). Note that eer common neighbor of and other than or is inside the ccle (,, ). hen, let be the first erte of the seqence that is a neighbor of. Since and hae at least to common neighbors, one of these ertices appears before in the seqence, so. On the other hand note that the erte ma be eqal to. Note that partner coples are defined for eer trianglation K 3. 2
e m Fig. 2. From e to m. Let l (resp. r ) be the trianglation indced b the ertices on and inside the ccle (,, ) (resp. (,, )). hen let e be the trianglation indced b the ertices on and otside the ccle (,,, ) (see Figre 1). Since / V ( r ) and / V ( l ), r and l hae less ertices than. his is not the case for e if (,, ) and (,, ) both delimit an inner-face of. In e, the ertices and (resp. and ) hae onl to common neighbors, and (resp. and ). So in e, the partner cople of (, ) is still (, ). We constrct m from e b deleting three edges,, and, and then merging and in a single erte (see Figre 2). Since and hae onl to common neighbors and in e, m is a ell defined trianglation, ithot loop or mltiple edges. Since e merged to ertices, m has less ertices than e. If m K 3, let (, ) be the partner cople of (, ) in m. Note that since and hae eactl to neighbors in e, and, the erte is adjacent to and not adjacent to in e. Using this decomposition, e proe the folloing theorem illstrated in Figre 3. heorem 4 Gien an trianglation = (V, E) and an triplet (,, ) of ertices on the eternal face, let (, ) be the partner cople of (, ). he graph = \{,, } has an F(,, 4)-coering b F 1, F 2 and F 3. If is defined, this is if K 3 these forests are sch that : - the edges of incident to are in F 1, - the edges of incident to are in F 2, - the edge is in F 3, - the edges of incident to strictl beteen and are in F 1, - the edges of incident to strictl beteen and are in F 2, and - the ertices, and are in distinct connected components of F i, for 1 i 3. Frthermore, the connected component of F 2 containing the erte onl contains and some ertices inside the ccle (,, ). Note that each of these forests has eactl 3 connected components. Indeed, an acclic graph on n ertices ith c connected components has n c edges and the graph has 3n 9 edges. For eample the forest F 2 has 2 connected 3
F F F 1 2 3 Fig. 3. he edge partition of heorem 1. components of one erte each, respectiel and, and a third connected component containing all the remaining ertices. We can etend this edgepartition of to b ptting for eample the edges and in F 1 and the edge in F 2. his partition clearl implies heorem 1. PROOF. his proof orks b indction on V (). he theorem clearl holds for K 3, so e consider the indction step of the proof. Gien a trianglation ith V () 4, consider the three trianglations m, l and r obtained b the decomposition of described before. Since m, l and r hae less ertices than, e can se the indction hpothesis. Let F 1, F 2 and F 3 be the three forests gien b the theorem for the trianglation m and the triplet (,, ). hese forests coer m = m \ {,, } and e se them to define the graphs F 1, F 2 and F 3 that coer e = e \ {,, }. - F i \ {, } F i - a F i if a and a F i - a F i if a, or, and a F i - F 1 - F 2 - F 3 hese forests erif the conditions of the theorem for the trianglation e and the triplet (,, ). - he edges incident to, or are clearl ell partitionned. - he graph F 1 is a forest. If F 1 had a ccle, this ccle shold either pass throgh or not. In the first case, this old impl that there is a path from to in F 1 m. In the second case, this old impl that there is a ccle in F 1 m. Both cases are impossible since the partition of m erifies the theorem. Similarl the ertices, and are in distinct connected components of F 1. If there as a path in F 1 linking to of these ertices this path shold either pass throgh or not. In the first case, this old impl that there is either a path from to or a ccle (passing throgh ) in F 1 m. In 4
e m Fig. 4. From m to e. the second case, this old impl that there is a path in F 1 m linking to of the ertices, or. Both cases are impossible since the partition of m erifies the theorem. - he graph F 2 is a forest. If F 2 had a ccle, this ccle shold either pass throgh or not. In the first case, this old impl that there is a path from to in F 2 m. In the second case, this old impl that there is a ccle in F 2 m. Both cases are impossible since the partition of m erifies the theorem. Similarl the ertices, and are in distinct connected components of F 2. If there as a path in F 2 linking to of these ertices this path shold either pass throgh or not. In the first case, this old impl that there is either a path from to or, or a ccle (passing throgh ) in F 2 m. In the second case, this old impl that there is a path in F 2 m linking to of the ertices, or. Both cases are impossible since the partition of m erifies the theorem. Frthermore, since there is no erte inside (,, ) and no edge incident to in F 2, the connected component of F 2 containing is as epected. - he graph F 3 is a forest. If F 3 had a ccle, this ccle shold either pass throgh or not. In the first case, this old impl that there is a ccle (passing throgh ) in F 3 m. In the second case, this old impl that there is a ccle in F 3 m. Both cases are impossible since the partition of m erifies the theorem. Similarl the ertices, and are in distinct connected components of F 3. If there as a path in F 3 linking to of these ertices this path shold either pass throgh or not. In the first case, this old impl that there is a path from to or in F 3 m. In the second case, this old impl that there is a path in F 3 m linking to of the ertices, or. Both cases are impossible since the partition of m erifies the theorem. Frthermore note that eer erte a V ( e ) \ {, } has as man incident edges in F 3 as in F 3. Since and hae respectiel one and to incident edges in F 3, F 3 has maimm degree at most for. For the rest of the proof it is important to remember that the theorem holds 5
l r e Fig. 5. From e, l and r to. for e in sch a that has degree to in F 3. he graph is the disjoint nion of e, l = l \{,, }, and r = r \{,, }. We constrct an edge-partition of into three forest F 1, F 2, and F 3, b partitionning each of e, l, and r into three forests. o do this, e appl the indction hpothesis to l according to the triplet (,, ). his means that the edges incident to (resp. ) in l belongs to F 1 (resp. F 2 ). Similarl, e appl the indction hpothesis to r according to the triplet (,, ). his means that the edges incident to (resp. ) in l belongs to F 1 (resp. F 2 ). We hae seen that the indction hpothesis holds for e according to the triplet (,, ), and e consider sch partition in hich the erte has degree to in F 3. his elds to a partition of into the three forests described in the theorem. - he edges incident to, or are clearl ell partitionned. - he graph F i, for an 1 i 3, is a forest. If F i old contain a ccle, since there is no sch ccle in e, l, or r, this ccle shold pass throgh l or r. his old impl that there is a path in F i l or F i r linking to of the ertices,,, and, hich is impossible according to the partitions of l or r. - he graph F i, for an 1 i 3, does not contain an path linking to of the ertices,, and. If F i old contain sch path, since there is no sch path in e, this path shold pass throgh l or r from to, or, or from to, hich is impossible according to the partitions of l or r. - Since the erte has no incident edges in F 2 e and F 2 r, and since there is no path from to or in F 2 l, the connected component of F 2 containing onl contains ans some ertices inside (,, ). - he graph F 3 is sch that, (F 3 ) 4. Indeed, has at most 2, 1, and 1 incident edges in F 2 e, F 2 l, and F 2 r ; and the other ertices hae as man incident edges in F 2 as in F 2 e, F 2 l, or F 2 r. his complete the proof of the theorem. 6
3 Planar graphs of bonded girth he reslts in [1, 2, 5, 6] are all proed sing discharging methods. We se this method for proing heorem 2 and heorem 3. his method consists roghl in shoing that a conter-eample H minimizing V (H) old be too dense (i.e. has too man edges per erte) to erif Eler s Formla. his formla sas that an connected planar graph G ith n ertices, m edges and f faces erifies m = n + f 2. Let s define a k-erte (resp. k-erte and k-erte) as a erte of degree k (resp. at most k and at least k). 3.1 Planar graphs ith girth g 6 Let H be a conter-eample of heorem 2 minimizing V (H). Lemma 5 he conter-eample H: (1) is connected, (2) has minimm degree δ(h) 2, and (3) does not contain an edge sch that deg() = 2 and deg() 5. PROOF. (1) If H as disconnected, one of its connected component old be a smaller conter-eample. (2) If H had a 1-erte, the graph H \ {} old hae girth g 6 and old hae an F(, 4)-coering b F 1 and F 2. Adding the incident edge of in F 1 e old obtain an F(, 4)-coering of H, hich is impossible. (3) Consider that H had an edge sch that deg() = 2 and deg() 5. Since H is minimal, the graph H\{} has an F(, 4)-coering b F 1 and F 2. We etend those forests to obtain an F(, 4)-coering of H. Let be the second neighbor of. If all the edges incident to in H\{} are in F 2 then let F 1 = F 1 {, } and F 2 = F 2. Else, has degree at most 3 in F 2 and let F 1 = F 1 {} and F 2 = F 2 {}. In both cases the forests F 1 and F 2 coer H, and (F 2) 4. Since H is not F(, 4)-coerable e hae a contradiction and H does not contain sch edge. We no se a discharging procedre on the ertices of H in order to estimate 2 E(H) / V (H). Let the intitial charge of the ertices be eqal to their degree, ch() = deg() for all V (H). hen, eer 6-erte gies charge 1 to its 2 neigbhors of degree 2. After this procedre the total charge of the graph is presered and all the ertices hae a final charge ch () 3. Indeed : If deg() = 2, then receies 1 2 and ch () = 2 + 2 1 = 3. 2 from each of its neighbors (Lemma 5.(3)) 7
If 3 deg() 5, then does not gie an charge, so ch () 3. If 6 deg(), then gies at most 1 2 to each of its neigbors, so ch () 6 6 1 2 = 3. So e hae that 2 E(H) = V (H) deg() = V (H) ch () 3 V (H). Let n, m and f denote respectiel the nmber of ertices, edges and faces in H. We kno that 2m 3n and since H has girth at least 6, each face is bonded b at least 6 edges and 2m 6f. Combining these to eqations e obtain that m n + f contradicting Eler s Formla. So H does not eist and heorem 2 holds. 3.2 Planar graphs ith girth g 7 Let H be a conter-eample of heorem 3 minimizing V (H). Lemma 6 he conter-eample H: (1) is connected, (2) has minimm degree δ(h) 2, (3) does not contain an edge sch that deg() = 2 and deg() 3, and (4) does not contain an 3-erte adjacent to three 3-ertices. PROOF. (1) If H as disconnected, one of its connected component old be a smaller conter-eample. (2) If H had a 1-erte, the graph H \ {} old hae girth g 6 and old hae an F(, 4)-coering b F 1 and F 2. Adding the incident edge of in F 1 e old obtain an F(, 4)-coering of H, hich is impossible. For the cases (3) and (4) e consider the graph H\{}. B minimalit of V (H), the graph H\{} has an F(, 4)-coering b F 1 and F 2. We consider a pair (F 1, F 2 ) maimizing the nmber of edges in F 1. his implies that eer 2-erte in H\{} has at most one incident edge in F 2. In case (3), let be the second neighbor of. Since has degree at most one in F 2, the forests F 1 {} and F 2 {} old be an F(, 4)-coering of H, hich is impossible. In case (4), let 1, 2 and 3 be the neighbors of. Since 1, 2 and 3 hae degree at most to in H\{}, the hae degree at most one in F 2. Since each connected component of F 2 contains at most to 1-ertices, to of the ertices 1, 2 and 3 are in distinct connected components of F 2, sa 1 and 2. In this case, the forests F 1 { 3 } and F 2 { 1, 2 } old be an F(, 4)-coering of H, hich is impossible. Since δ(h) 2, e distingish 6 tpes of edges in H : 8
(a) For eer 2-erte, let one of its incident edges be an a-edge and the other one be an a-edge. Let s distingish 2 tpes of 3-ertices. An isolated 3-erte has no 3-erte in its neighborhood. he rest of the 3-ertices are linked 3-ertices, this means adjacent to at least one 3-erte. (b) For eer isolated 3-erte, let one of its incident edges be a b-edge and the to remaining ones be b-edges. (c) We consider the sbgraph K of H indced b the linked 3-ertices. his sbgraph K is sch that (K) 2 (b Lemma 6.(4)) and δ(k) 1 (b definition of linked 3-ertices). Let C E(K) be the smallest set of edges in K sch that all the linked 3-ertices hae an incident edge in C. he minimalit of C implies that in each connected component of K (a ccle or a path), there is at most one erte ith to incident edges in C. he edges of C are the c-edges and all the edges of H (not jst K) adjacent to a c-edge are c-edges. It is clear gien Lemma 6 that the sets of a-, b-, c-, a-, b- and c-edges, respectiel A, B, C, A, B and C, are pairise disjoint. No e transform H into another graph H b contracting the a-, b- and c-edges. Since eer 2-erte (resp. 3-erte) is adjacent to a 4-erte (resp. 3-erte) b an a-edge (resp. b- or c-edge), and since it has at most one (resp. to) incident a-edge (resp. b- or c-edges), there is no more ertices of degree less than 4 in H. Lemma 7 he graph H is connected and after the transformation eer ccle C in H becomes a ccle C in H. (1) If C has length 7 and if all its ertices are 3-ertices, then the ccle C has length 3 and contains a 5-erte. (2) Else, the ccle C has length l(c ) 4. PROOF. It is clear that, b contracting edges, a graph remains connected. For the ccles e distingish the to cases. In case (1), since C has 4 c- edges and 3 c-edges, the ccle C has length 3 and the to consectie c-edges prodce a 5-erte. In case (2), the ccle C contains at least one 4-erte (case (2.1)) or contains onl 3-ertices and has length l 8 (case(2.2)). In case (2.1), consider an path P = ( 0, 1,..., k ) C linking to 4- ertices, 0 and k, and going throgh 3-ertices. Actall this path ma be a ccle if 0 = k. Claim 8 here is at least as man a-edges (resp. b-edges and c-edges) in P than a-edges (resp. b-edges and c-edges). 9
Indeed : (-) If P is jst an edge linking to 4-ertices, then this edge is not an a-, b- or c-edge. (a) If P goes throgh a 2-erte, then P has length 2 and contains eactl one a-edge and one a-edge. (b) If P goes throgh an isolated 3-erte, then P has length 2 and contains at most one b-edge and at least one b-edge. (c) If P goes throgh (k 1) 3-ertices, then P contains k 1 c-edges and the remaining k k 1 2 edges are c-edges. his claim implies that at most half of the edges in C are contracted. Since l(c) 7 this implies that C has length l 4. In case (2.2), the ccle C has length l 8 and contains l 2 c-edges and the remaining l 2 edges are c-edges Since l 2 4 hen l 8, e hae l(c ) 4 and the lemma holds. 2 Let n 4 and n 5 be the nmber of 4-ertices and 5-ertices in H. Let c 3 be the nmber of ccles of length 3 in H. Note that all the ccles of length 3 in H contain a 5-erte. Since these ccles of length 3 in H come from ccles of 3-ertices in H, Lemma 6.(4) implies that these ccles of length 3 are erte disjoint. his implies that n 5 c 3. Let f 3 and f 4 be the nmber of faces of length respectiel l = 3 and l 4 in H. Since c 3 f 3, e hae n 5 f 3. No, let n, m and f be the nmber of ertices, edges and faces in H. It is clear that n = n 4 +n 5 and f = f 3 +f 4. Since the edges hae to end points and are incident to at most to faces, e hae : 2m 4n 4 + 5n 5 = 4n + n 5 4n + f 3 2m 3f 3 + 4f 4 = 4f f 3 Sming these to eqations e obtain that m n + f contradicting Eler s Formla. So H and H do not eist and heorem 3 holds. 4 Perspecties In [3] Colin de Verdière introdced a graph parameter µ. For a graph G this parameter is defined b spectral properties of matrices associated to G. his parameter is sch that : - µ(g) 1 iff G is a forest of paths. - µ(g) 2 iff G is an oterplanar graph. 10
- µ(g) 3 iff G is a planar graph. Since forests of paths, oterplanar graphs [2], and planar graphs are respectiel F(2)-, F(, 3)-, and F(,, 4)-coerable e conjectre the folloing. Conjectre 9 Eer graph G has an edge partition into µ(g) forests, one haing maimm degree µ(g) + 1. A eaker reslt old be that eer graph G is (µ(g) 1, µ(g)+1)-coerable. his reslt old be sharp, indeed: heorem 10 For an integer k 1 there is a graph G ith µ(g) = k that is not (k 1, k)-coerable. PROOF. It is kno for k 3, so consider that k 4. For an pair of positie integers (k, l) ith k 4 and l 0 e define the graph G l k. Let G0 k = K k+1. For l > 0 e constrct the graph G l k from Gl 1 k b adding, for each cop of K k in G l 1 k that contain a k-erte a ne erte adjacent to the ertices in this cop of K k. According to [7] e hae µ(g l k ) = k for an k and l. Claim 11 For an l 1, the graph G l k has (k + 1)k l 1 k-ertices that form an independant set and (k + 1) ( 1 + l 2 i=0 k i) > k-ertices. Frthermore, this graph has k(k+1) +(k +1) ( l 1 2 i=1 ki) edges linking to > k-ertices and (k +1)k l edges linking a k-erte and a > k-erte. Indeed, it is clear for l = 1 and for the indction e jst note that each of the (k + 1)k l 2 k-ertices in G l 1 k belongs to k copies of K k. Since these ertices form an independant set there is k (k + 1)k l 2 copies of K k in G l 1 k that contain a erte of degree k. So G l k has (k + 1)kl 1 ne ertices of degree k and all the ertices that ere in G l 1 k hae no degree more than k, and there are (k + 1) ( 1 + l 2 i=0 ki) sch ertices. Frthermore, since eer k-erte of G l k is incident to > k-ertices, these k-ertices clearl form an independant set. For the nmber of edges, it is clear that from G l 1 k to G l k e add k ne edges per ne erte (of degree k) and that eer edge present in G l 1 k link to > k-ertices in G l k. We consider no the folloing theorem of Balogh et al. [2]. heorem 12 For eer (t, D)-coerable graph G and an to disjoint sbsets A and B of V (G), f t (A) + e(a, B) D A + t( A + B 1) here e(x, Y ) denotes the nmber of edges of G ith one end in X and the other in Y, and here f t (A) = e(a, A) if e(a, A) t( A 1), and f t (A) = 11
2e(A, A) t( A 1) otherise. For an k 4 consider the graph G 3 k, let A be the set of > k-ertices and let B be the set of k-ertices. his theorem sas that if G 3 k as (k 1, k)-coerable e shold hae f k 1 (A) + e(a, B) k A + (k 1)( A + B 1) Note that according to Claim 11 A = (k + 1)(k + 2) and e(a, A) = k(k + 1)(k+3/2), so e hae e(a, A) > (k 1)( A 1). his implies that f k 1 (A) = 2e(A, A) (k 1)( A 1) = 1 2 (k2 +7k+2). hs if G 3 k as (k 1, k)-coerable e shold hae 1 2 (k2 + 7k + 2) + (k + 1)k 3 k(k + 1)(k + 2)+ (k 1)((k + 1)(2 + k + k 2 ) 1), hich is eqialent to 2 + 6k 6k 2 k 3 + k 4 0, and hich does not hold for k 4. hs G 3 k is not (k 1, k)-coerable for k 4 and this complete the proof of the theorem. Another interesting qestion concerns the conseqences of heorem 1. Since the forests of maimm degree for are coerable b to linear forests or b to star forests ith maimm degree three e hae the folloing corollar. Corollar 13 Planar graphs are coerable b : - 6 star forests, to of them haing maimm degree at most three. - 2 forests and 2 linear forests. Planar graphs ith girth g 6 are coerable b : - 4 star forests, to of them haing maimm degree at most three. - 1 forest and 2 linear forests. We hae seen that heorem 1 is optimal, e onder if it is also the case for heorem 2, heorem 3 and for this corollar. References [1] A. Bassa, J. Brns, J.Campbell, A. Deshpande, J. Farle, M. Halse, S. Michalakis, P.-O.Persson, P. Plask, L. Rademacher, A. Riehl, M. Rios, J. Samel, B. enner, A. Vijaasarat, L. Zhao, and D.J. Kleitman. 12
Partitioning a Planar Graph of Girth en into a Forest and a Matching manscript (2004). [2] J. Balogh, M. Kochol, A. Plhár and X. Y. Coering planar graphs ith forests, J. Combin. heor B. 94 (2005), 147-158. [3] Y. Colin de Verdière, Sr n noel inariant des graphes et n critère de planarité J. Combin. heor Ser. B 50 (1990) 1, 11 21. [4] D. Gonçales. Caterpillar arboricit of planar graphs, Discrete Math. to appear. [5] D.J. Gan, X. Zh, Game chromatic nmber of oterplanar graphs J. Graph heor 30 (1999) 1, 67 70. [6] W. He, X. Ho, K-W Lih, J. Shao, W. Wang and X. Zh, Edge-partitions of planar graphs and their game coloring nmbers J. Graph heor 41 (2002) 4, 307 317. [7] H. an der Holst, L. Loász and A. Schrijer On the inariance of Colin de Verdière s graph parameter nder cliqe sms Linear Algebra Appl. 226/228 (1995), 509 517. [8] C.St.J.A. Nash-Williams. Decompositions of finite graphs into forests, J. London Math. Soc. 39 (1964), 12. 13