More on the Linear k-arboricity of Regular Graphs R. E. L. Aldred Department of Mathematics and Statistics University of Otago P.O. Box 56, Dunedin Ne

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1 More on the Lnear k-arborcty of Regular Graphs R E L Aldred Department of Mathematcs and Statstcs Unversty of Otago PO Box 56, Dunedn New Zealand Ncholas C Wormald Department of Mathematcs Unversty of Melbourne Parkvlle, VIC 3052 Australa Abstract Bermond et al [2] conjectured that the edge set of a cubc graph G can be parttoned nto two lnear k-forests, that s to say two forests whose connected components are paths of length at most k, for all k 5 That the statement s vald for all k 18 was shown n [5] by Jackson and Wormald Here we mprove ths bound to k 7 f 0 (G) =3; 9 otherwse The result s also extended to d-regular graphs for d > 3, at the expense of ncreasng the number of forests to d, 1 All graphs consdered wll be nte We shall refer to graphs whch may contan loops or multple edges as multgraphs and reserve the term graph for those whch do not A lnear forest s a forest each of whose components 1

2 are paths The lnear arborcty of a graph G, dened by Harary [4], s the mnmum number of lnear forests requred to partton E(G) and s denoted by la(g) It was shown by Akyama, Exoo and Harary [1] that la(g) = 2 when G s cubc A lnear k-forest s a forest consstng of paths of length at most k The lnear k-arborcty of G, ntroduced by Bermond et al [2], s the mnmum number of lnear k-forests requred to partton E(G), and s denoted by la k (G) When such a partton of E(G) has been mposed, we say that G has been factored nto lnear k-forests We refer to each lnear forest n such a partton of E(G) as a factor of G and the partton tself s called a factorzaton It s conjectured n [2] that f G s cubc then la 5 (G) 2 A partal result s obtaned by Delamarre et al [3] who show that la k (G) 2 when k 1 jv (G)j 4 Jackson and Wormald [5] mproved ths result when 2 jv (G)j 36, showng that, for k 18, an nteger, la k (G) =2 Here we shall mprove onths further to show the followng Theorem Let G be a cubc graph and let k be an nteger Then la k (G) =2 for all k 7 f 0 (G) =3; 9 otherwse Before we prove the theorem, we shall present some prelmnary results whch ndcate some restrctons we can demand of factorzatons of cubc graphs These wll be most useful n the proof of the theorem For our rst result we ntroduce the followng termnology An odd lnear forest s a lnear forest n whch each component s a path of odd length Also, we use 0 (G) to denote the chromatc ndex of G (e the mnmum number of colours requred to colour the edges of G so that no two edges of the same colour are ncdent wth the same vertex) Lemma Let G be a cubc graph Then G can be factored nto two odd lnear forests f and only f 0 (G) =3 Proof Suppose rst that G s a cubc graph wth a factorzaton, (F 1 ;F 2 ) nto odd lnear forests F 1 and F 2 Colour the edges of the paths n F 1 alternately red and blue so that each path n F 1 has ts rst and last edges coloured red Smlarly, colour the edges of the paths n F 2 alternately green and blue so that each path n F 2 has ts rst and last edges coloured green Ths yelds a proper 3-edge-colourng of G gvng 0 (G) =3 2

3 Conversely, let us suppose that 0 (G) = 3 and that we have a proper 3-edge-colourng mposed on G usng the colours blue, green and red Let F 0 1 and F 0 2 be factors of G nduced by the blue and green edges and by the red edges respectvely Thus F 0 1 conssts of dsjont even cycles, whle F 0 2 s a set of dsjont edges coverng the vertces of G Form new factors F 1 and F 2 where F 2 s the subgraph of G nduced by the edges n F 0 2 together wth at most one edge from each cycle n F 0 1 chosen so that F 2 s acyclc and such that the paths n F 2 have maxmum possble total length The factor F 1 conssts of F 0 1 wth the edges n F 2 removed Clam: F 2 contans an edge from each cycle n F 0 1 To see ths, assume to the contrary that there s a cycle C = v 1 v 2 :::v k v 1 n F 1 Then v 1 and v 2 are both ends of the one path n F 2, whle v 2 and v 3 are both ends of the one path n F 2 (by the maxmalty of total length of paths n F 2 ) Snce G s a graph, v 1 6= v 3 and no path has three ends so the clam follows Thus each path n F 1 has odd length and each path n F 2 has odd length (these paths must begn and end wth red edges and have every second edge red throughout ther lengths) Corollary Let G be an r-regular graph If 0 (G) = r, 1, then G can be factored nto r, 1 odd lnear forests Proof Snce 0 (G) =r, the edges of G can be parttoned nto r 1-factors choose r, 3 of these to form factors n our factorzaton, the remanng 3 1-factors nducng a cubc spannng subgraph of G whch has chromatc ndex 3 By the lemma, ths subgraph can be factored nto two odd lnear forests, completng the desred factorzaton and the proof of the corollary Proposton Let G be a cubc graph Then there s a factorzaton of G nto two lnear forests, F 1 and F 2 such that no component n F somorphc to P 3 has ts end vertces adjacent n G va an nternal edge of a path n F j, where f; jg = f1; 2g (Such a factorzaton s to have the spread-p 3 property) Proof The proposton s certanly true for G = K 4 So assume that G s a cubc graph of smallest order whch does not admt a factorzaton wth the spread-p 3 property Clearly, G cannot be trangle-free But f we have a trangle T, wth each edge belongng to precsely one trangle, then contractng T to a vertex yelds a cubc graph, G 0, of smaller order than than G By the mnmaltyofg, G 0 admts a factorzaton wth the spread-p 3 property Ths 3

4 factorzaton easly lfts to a factorzaton of G wth the spread-p 3 property Consequently, we may assume that all trangles n G are pared By [1] we can let G be factored nto two lnear forests Assume ths s done so that there are as few unspread-p 3 's as possble (By unspread-p 3 we mean a path of length two n one factor whose endvertces are adjacentngva an nternal edge n the other factor) As G was chosen not to admt a factorzaton wth the spread-p 3 property, we must have an unspread-p 3 n one of the pared trangles Fgure 1 ndcates how ths factorzaton may be altered to reduce the number of unspread-p 3 's Ths contradcton establshes the result,,, - Fgure 1 Proof of Theorem Dene F = F(G) to be the set of (F 1 ;F 2 ) such that F 1 and F 2 form a factorzaton of G wth the spread-p 3 property Then F s nonempty by the proposton For (F 1 ;F 2 ) 2 F, let l(f 1 ;F 2 ) denote the length of the longest path n F 1 or F 2 Let t be the mnmum number such that there exsts (F 1 ;F 2 ) 2F wth l(f 1 ;F 2 )=t Choose (F 1 ;F 2 ) 2 F wth l(f 1 ;F 2 ) = t and such that, subject to ths condton, the total number m(f 1 ;F 2 ), of paths of length t n F 1 and F 2 s as small as possble By symmetry, we may assume that F 1 has a path P of length t Assume t 10 We wll use the structure of ths factorzaton to produce dstnct edges, e of G for each nteger 0, thereby contradctng the nteness of G To assst n ths process we dene an -mprovement Gven path sets S 0 ;:::;S, dene R = R (S 0 ;:::;S ) to be the set of all edges of G whch are ether contaned n or ncdent wth any of the paths n S 0 ;:::;S Gven (F 1 ;F 2 ) 2Fand path sets S 0 ;:::;S, we say that (F 0 1 ;F0 2) 2Fs an -mprovement of (F 1 ;F 2 ) f there exsts an edge e of a path n S such that () all edges n R n e are n F 0 1 they are n F 1, 4

5 () e s n F 0 1 t s not n F 1, and () each path n F 0 j of length at least t s also a path n F j, j = 1 and 2 We now descrbe the algorthm for producng the edges e (referred to as growth edges) The algorthm proceeds through steps of choosng a growth edge and \growng" one or two paths from ts endvertces The rst path \grown" s P Certan vertces become \poson" and later \kll" the \lve" edges, whch are the potental growth edges Intally we set: S 0 := fp g (denotng the set of one or two paths just \grown"); T 0 := a maxmum set of ndependent nternal edges n P excludng those thrd from ether end (\lve" edges, from whch t s possble to choose growth edges); W 0 := fend vertces of P g (\poson" vertces) Note that when t 5, jt 0 j = d t,2 2 e So for t 7we have have jt 0j > jw 0 j For >0 carry out the followng three steps 1 Choose e,1 2 T,1 2 Repeat (a) and (b) consecutvely untl (b) requres no acton because ts condton fals: (a) Dene S to be the set of paths whch termnate at the endvertces of e,1 (b) If there s an -mprovement (F 0 1 ;F0 2)of(F 1 ;F 2 ) then reset (F 1 ;F 2 ) to be equal to (F 0 1 ;F0 2) 3 Determne the sets T and W by the followng method Set: V := f endvertces of paths n S gnfendvertces of e,1g; 5

6 N := a maxmum set of ndependent nternal edges of paths n S other than those thrd along from a vertex n V ; T 0 := (T,1nfe,1g) [ N (addng edges n N to the set of lve edges, and deletng e,1); W 0 := W,1 [ V (V contans new poson vertces); T := T 0 nfe 2 T 0 ncdent wth a vertex n W 0 g (these edges are posoned); W := W 0 nfw 2 W 0 ncdent wth and edge n T 0 g (after a vertex posons an edge, t s no longer poson) Step 2 must eventually termnate because the redenton of (F 1 ;F 2 ) n 2(b) leads to a strctly smaller total length of paths n the set S when 2(a) s next performed Denote the value of (F 1 ;F 2 ) at the end of the th teraton of the two steps above by (F ;1;F ;2) Then for any j >the set S retans ts property of beng a set of maxmal paths n F j;1 or F j;2, because the redenton of (F 1 ;F 2 ) does not dsturb any edges n R j,1 Thus, (F j;1;f j;2) cannot have any -mprovement for <j Note also that prospectve growth edges n N are posoned by vertces n W 0 precsely when such an edge shares an endvertex wth a path n some S j ;j, 1 In ths way, the paths n S cannot be ncluded n S j ;j, 1 Thus an edge e h 2 T h chosen n Step 1 cannot be contaned n T h 0 whenever h 0 >h Consequently, havng chosen e,1 2 T,1 for Step 1, we have one of two possbltes at the completon of Step 2 () There s a sngle path, P 0,nS jonng the endvertces of e,1 Snce we are workng wth factorzatons wth the spread-p 3 property, P 0 must have length at least 3 As we work through Step 3 n ths case, V = ; and we have at least one edge n N Each edge of T 0 \klled" by lve vertces n W 0 results n one vertex from W,1 not beng retaned n W and, n ths case, no new vertces wll be added to W,1 when formng W (snce V = ;) Thus jt j ncreases by at least as much as jw j () There are two paths P; ~ ^P 2 S termnatng at the endvertces of e,1 Suppose that l( P ~ )+l( ^P ) t, 2 Then e,1 cannot be n S 0 = P (f t 6

7 were, then swappng e,1 between the approprate factors would produce a factorzaton n F n whch the total number of paths of length at least t s less than n our orgnally chosen factorzaton; a contradcton) Thus e,1 s n some path n S j ; 1 j < But now swappng e,1 between the approprate factors wll yeld a j-mprovement for some j < (The nternal edges thrd from a "lve" end of a path n S j ; 0 j, 1 have been excluded from T,1 to ensure that swappng e,1 between factors cannot result n an unspread-p 3, producng a factorzaton not n F) Thus we have l( P ~ )+l( ^P ) t, 1 Workng through Step 3 n ths case, t can be checked that N contans at least three edges, as t 10 As before, each edge of T 0 \posoned" by a vertex n W 0 results n one vertex from W 0 not beng retaned n W There are at most two new vertces added to W,1 to form W 0 and at least three new edges added to T,1 to obtan T 0 Once agan, jt j ncreases by at least as much as jw j In both cases we see by nducton that jt j (the number of lve edges) exceeds jw j (the number of poson vertces) Thus we may contnue ndentely, removng a derent edge from T at each teraton Ths contradcts the nteness of G Next assume that 0 (G) = 3 By the Lemma, G can be factored nto two odd lnear forests Workng through the proof above restrctng F to the set of factorzatons of G nto two odd lnear forests, we make the followng mnor modcatons T 0 := every second edge n P ; N := every second edge n paths n S Now, everythng follows as before snce n case () above, the paths ~ P; ^P are restrcted to be of odd length Consequently, the nteness of G s contradcted f t 9, as all paths are of odd length and thus t, the length of a longest path n ether factor, must also be odd The result follows Fnally we note that our Theorem can easly be extended to graphs of maxmum degree three snce any such graph can be embedded as a subgraph of a cubc graph Usng ths we deduce the followng: 7

8 Corollary Let G be a 4-regular graph and k 9 be an nteger la k (G) =3 Then Proof Usng [7] we may choose a 2-factor F n G Clearly F has a spannng k-lnear forest D Snce H = G,E(D) has maxmum degree three, t follows from the above-mentoned extenson of our Theorem that la k (H) =2 Thus la k (G) =3 In [6], Lndquester and Wormald consdered the followng varaton of lnear arborcty for r-regular graphs An r-regular graph G s sad to be (l; k)-lnear arborc f t can be factored nto l lnear k-forests In ths settng, our results enable us to prove the followng Corollary Let G be an r-regular graph, 3, and let k be an nteger Then G s (r, 1;k)-lnear arborc for all k 7 f 0 (G) =r; 9 otherwse Proof If 0 (G) = r, then we may factor G nto r 1-factors Choose r, 3 of these 1-factors and delete them leavng a cubc spannng subgraph of G whch s 3-edge-colourable By our theorem, ths subgraph s (2; 7)-lnear arborc and thus G s (r, 1; 7)-lnear arborc as requred When 0 (G) 6= r, we proceed by nducton on r Our theorem and the above corollary show that ths s true for r =3; 4 Assume that the statement s true for all r r 0 and let G be an r regular graph If r 0 +1s even, then G has a 2-factor, F, say The graph G, F s regular of degree r 0, 1 and so, by nducton, may be factored nto r 0,2 lnear 9-forests The desred factorzaton s completed by decomposng F nto two lnear 2-forests n the obvous way So assume that r 0 +1 s odd Let M be a maxmum matchng n G Then G, M has vertces of degrees r 0 and r 0 + 1, and the vertces of degree r form an ndependent set, B, say The subgraph of G, M nduced by B [ N(B) wth edges between vertces n N(B) deleted s bpartte and, by Hall's theorem and consderatons of degrees, contans a matchng, M 0, whch covers the vertces n B Thus G 0 = G, (M [ M 0 ) s bregular wth degrees r 0 and r 0, 1 If H 0 s an somorphc copy ofg 0 and G 00 s the graph obtaned from G 0 and H 0 by addng a matchng between the vertces of degree r 0, 1 n G 0 and the vertces of degree r 0, 1 n H 0, then G 00 s r 0 -regular and, by our nductve hypothess, contans a factorzaton nto r 0, 1 lnear 9-forests 8

9 Restrctng these factors to the vertces n G together wth M [ M 0 gves the desred factorzaton of G nto r 0 lnear 9-forests (M [ M 0 s a lnear forest wth components of sze 1 and 2) References [1] J Akyama, G Exoo and F Harary, Coverng and packng n graphs III, cyclc and acyclc nvarants, Math Slovaca, 30 (1980) [2] JC Bermond, JL Fouquet, M Habb and B Peroche, On lnear k- arborcty, Dscrete Math, 52 (1984) [3] D Delamarre, JL Fouquet, H Thuller and B Vrot, Lnear k- arborcty of cubc graphs, preprnt Magyar Tud [4] F Harary, Coverng and packng n graphs I, Ann New York Acad Sc, 175 (1970) [5] Bll Jackson and Ncholas C Wormald, On the lnear arborcty of cubc graphs, to appear n Dscrete Math [6] T Lndquester and Ncholas C Wormald, Lnear arborcty of r-regular graphs, submtted [7] J Petersen, De Theore der regularen Graphen, Acta Math, 15 (1891),

the nber of vertces n the graph. spannng tree T beng part of a par of maxmally dstant trees s called extremal. Extremal trees are useful n the mxed an

the nber of vertces n the graph. spannng tree T beng part of a par of maxmally dstant trees s called extremal. Extremal trees are useful n the mxed an On Central Spannng Trees of a Graph S. Bezrukov Unverstat-GH Paderborn FB Mathematk/Informatk Furstenallee 11 D{33102 Paderborn F. Kaderal, W. Poguntke FernUnverstat Hagen LG Kommunkatonssysteme Bergscher