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On the Linear K-Arboricity of Regular Graphs More on the Linear k-arb oricity of Regular Graphs R. E. L. Aldred Department of Mathematics and Statistics University of Otago P.O. Box 56, Dunedin New Zealand Nicholas C. Wormald Department of Mathematics University of Melb ourne Parkville, VIC 3052 Australia Abstract Bermond et al. [2] conjectured that the edge set of a cubic graph G can b e partitioned into two linear k -forests, that is to saytwo forests whose connected comp onents are paths of length at most k , for all k 5. That the statement is valid for all k 18 was shown in [5 ] by Jackson and Wormald. Here we improve this b ound to 0 7 if G=3; k 9 otherwise. The result is also extended to d-regular graphs for d > 3, at the exp ense of increasing the numb er of forests to d 1. All graphs considered will b e nite. We shall refer to graphs which may contain lo ops or multiple edges as multigraphs and reserve the term graph for those which do not. A linear forest is a forest each of whose comp onents 1 are paths. The linear arboricity of a graph G, de ned by Harary [4], is the minimum numb er of linear forests required to partition E G and is denoted by laG. It was shown by Akiyama, Exo o and Harary [1] that laG = 2 when G is cubic. A linear k -forest is a forest consisting of paths of length at most k . The linear k -arboricity of G, intro duced by Bermond et al. [2], is the minimum number of linear k -forests required to partition E G, and is denoted by la G. When such a partition of E G has b een imp osed, we k say that G has b een factored into linear k -forests. We refer to each linear forest in such a partition of E G as a factor of G and the partition itself is called a factorization. It is conjectured in [2] that if G is cubic then la G 2. A partial 5 result is obtained by Delamarre et al. [3] who show that la G 2 when k 1 k jV Gj 4. Jackson and Wormald [5] improved this result when 2 jV Gj 36, showing that, for k 18, an integer, la G= 2. Here we shall k improveon this further to show the following. Theorem. Let G b e a cubic graph and let k b e an integer. Then la G= 2 k for all 0 7 if G= 3; k 9 otherwise. Before we prove the theorem, we shall present some preliminary results which indicate some restrictions we can demand of factorizations of cubic graphs. These will b e most useful in the pro of of the theorem. For our rst result weintro duce the following terminology. An odd linear forest is a linear forest in which each comp onent is a path of odd length. 0 Also, we use G to denote the chromatic index of G i.e. the minimum number of colours required to colour the edges of G so that no two edges of the same colour are incident with the same vertex. Lemma. Let G be a cubic graph. Then G can be factored into two odd 0 linear forests if and only if G=3. Pro of. Supp ose rst that G is a cubic graph with a factorization, F ;F 1 2 into odd linear forests F and F . Colour the edges of the paths in F al- 1 2 1 ternately red and blue so that each path in F has its rst and last edges 1 coloured red. Similarly, colour the edges of the paths in F alternately green 2 and blue so that each path in F has its rst and last edges coloured green. 2 0 This yields a prop er 3-edge-colouring of G giving G= 3. 2 0 Conversely, let us supp ose that G = 3 and that we have a prop er 3-edge-colouring imp osed on G using the colours blue, green and red. Let 0 0 F and F be factors of G induced by the blue and green edges and by the 1 2 0 0 red edges resp ectively. Thus F consists of disjoint even cycles, while F is 1 2 a set of disjoint edges covering the vertices of G. Form new factors F and 1 0 F where F is the subgraph of G induced by the edges in F together with 2 2 2 0 at most one edge from each cycle in F chosen so that F is acyclic and such 2 1 that the paths in F have maximum p ossible total length. The factor F 2 1 0 consists of F with the edges in F removed. 2 1 0 Claim: F contains an edge from each cycle in F . To see this, assume 2 1 to the contrary that there is a cycle C = v v :::v v in F . Then v and v 1 2 k 1 1 1 2 are b oth ends of the one path in F , while v and v are b oth ends of the one 2 2 3 path in F by the maximality of total length of paths in F . Since G is a 2 2 graph, v 6= v and no path has three ends so the claim follows. 1 3 Thus each path in F has o dd length and each path in F has o dd length 1 2 these paths must b egin and end with red edges and haveevery second edge red throughout their lengths. 0 Corollary. Let G be an r -regular graph. If G = r 1, then G can be factored into r 1 odd linear forests. 0 Pro of. Since G=r , the edges of G can be partitioned into r 1-factors. cho ose r 3 of these to form factors in our factorization, the remaining 3 1-factors inducing a cubic spanning subgraph of G which has chromatic index 3. By the lemma, this subgraph can be factored into two o dd linear forests, completing the desired factorization and the pro of of the corollary. Prop osition. Let G be a cubic graph. Then there is a factorization of G into two linear forests, F and F such that no comp onent in F isomorphic 1 2 i to P has its end vertices adjacent in G via an internal edge of a path in 3 F , where fi; j g = f1; 2g. Such a factorization is to have the spread-P j 3 prop erty. Pro of. The prop osition is certainly true for G = K . So assume that G is a 4 cubic graph of smallest order which do es not admit a factorization with the spread-P prop erty. Clearly, G cannot b e triangle-free. But if we have a tri- 3 angle T , with each edge b elonging to precisely one triangle, then contracting 0 T to a vertex yields a cubic graph, G , of smaller order than than G. By the 0 minimalityof G, G admits a factorization with the spread-P prop erty. This 3 3 factorization easily lifts to a factorization of G with the spread-P prop erty. 3 Consequently, we may assume that all triangles in G are paired. By [1] we can let G b e factored into two linear forests. Assume this is done so that there are as few unspread-P 's as p ossible. By unspread-P we mean a path 3 3 of length two in one factor whose endvertices are adjacentinG via an internal edge in the other factor. As G was chosen not to admit a factorization with the spread-P prop erty, we must have an unspread-P in one of the paired 3 3 triangles. Figure 1 indicates how this factorization may b e altered to reduce the number of unspread-P 's. This contradiction establishes the result. 3 . @ @ . @ . @ . @ @ . - . @ @ . . @ @ . @ @ . Figure 1. Pro of of Theorem. De ne F = F G to be the set of F ;F such that 1 2 F and F form a factorization of G with the spread-P prop erty. Then F 1 2 3 is nonempty by the prop osition. For F ;F 2 F , let l F ;F denote the 1 2 1 2 length of the longest path in F or F . Let t be the minimum number such 1 2 that there exists F ;F 2F with l F ;F = t. 1 2 1 2 Cho ose F ;F 2 F with l F ;F = t and such that, sub ject to this 1 2 1 2 condition, the total number mF ;F , of paths of length t in F and F is 1 2 1 2 as small as p ossible. By symmetry, we may assume that F has a path P of 1 length t. Assume t 10. We will use the structure of this factorization to pro duce distinct edges, e of G for each integer i 0, thereby contradicting the i niteness of G. To assist in this pro cess we de ne an i-improvement. Given path sets S ;:::;S , de ne R = R S ;:::;S to be the set of all 0 i i i 0 i edges of G which are either contained in or incident with any of the paths in S ;:::;S . 0 i 0 0 Given F ;F 2F and path sets S ;:::;S , we say that F ;F 2F is 1 2 0 i 1 2 an i-improvement of F ;F if there exists an edge e of a path in S such 1 2 i that 0 i they are in F , i all edges in R n e are in F 1 i 1 4 0 ii e is in F i it is not in F , and 1 1 0 iii each path in F of length at least t is also a path in F , j = 1 and 2.
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