More on the linear k-arboricity of regular graphs

Bermond et al. [5] conjectured that the edge set of a cubic graph G can be partitioned into two linear k-forests, that is to say two forests whose connected components are paths of length at most k, for all k ;::: 5. That the statement is valid for all k ;::: 18 was shown in [8] by Jackson and Wormald. Here we improve this bound to k > {7 if X'( G) = 3; 9 otherwise. The result is also extended to d-regular graphs for d > 3, at the expense of increasing the number of forests to d 1. All graphs considered will be finite. We shall refer to graphs which may contain loops or multiple edges as multigraphs and reserve the term graph for those which do not. A linear forest is a forest each of whose components is a path. The linear arboricity of a graph G, defined by Harary [7], is the minimum number of linear forests required to partition E(G) and is denoted by la(G). It was shown by Akiyama, Exoo and Harary [1] that la( G) = 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, introduced by Bermond et al. [5], is the minimum number oflinear k-forests required to partition E(G), and is denoted by lak:( G). When such a partition of E( G) has been imposed, we say that G has been 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 [5] that if G is cubic then la5( G) :::; 2. A partial result is obtained by Delamarre et al. [6] who show that lak:(G) :::; 2 when k ;::: ~IV(G)I ;::: 4. Australasian Journal of Combinatorics 18(1998), pp.97-104 Jackson and Wormald [8] improved this result when IV(G)I 2:: 36, showing that, for k ~ 18, an integer, lak(G) = 2. Here we shall improve on this further to show the following. Theorem 1. Let G be a cubic graph and let k be an integer. Then lak( G) = 2 for all k > {7 if X' (G) = 3; 9 otherwise. In [9], Lindquester and Wormald considered the following variation of linear arboricity for r-regular graphs. An r-regular graph G is said to be (I, k )-linear arborific if it can be factored into l linear k-forests. In this setting, we are able to prove the following. Theorem 2. Let G be an r-regular graph, r ~ 3, and let k be an integer. Then G is (r -1, k)-linear arborific for all k > {7 if X' (G) = r; 9 otherwise. While Theorems 1 and 2 are proved here for finite graphs, the results also apply for infinite graphs using a standard method (by Tihonov's theorem see for example the application on page 57 of [2]). Before we prove the theorems, we shall present some preliminary results which indicate some restrictions we can demand of factorizations of cubic graphs. These will be most useful in the proof of the theorem. For our first result we introduce the following terminology. An odd linear forest is a linear forest in which each component is a path of odd length. Also, we use X'(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 linear forests if and only ifX'(G) = 3. Proof. Suppose first that G is a cubic graph with a factorization, (Fl' F2 ) into odd linear forests Fl and F2 • Colour the edges of the paths in Fl alternately red and blue so that each path in Fl has its first and last edges coloured red. Similarly, colour the edges of the paths in F2 alternately green and blue so that each path in F2 has its first and last edges coloured green. This yields a proper 3-edge-colouring of G giving X'(G) = 3. Conversely, let us suppose that X'( G) = 3 and that we have a proper 3-edgecolouring imposed on G using the colours blue, green and red. Let F{ and F~ be factors of G induced by the blue and green edges and by the red edges respectively. Thus F{ consists of disjoint even cycles, while F~ is a set of disjoint edges covering the vertices of G. Form new factors Fl and F2 where F2 is the subgraph of G induced by the edges in F~ together with at most one edge from each cycle in F{ chosen so that F2 is acyclic and such that the paths in F2 have maximum possible total length. The factor Fl consists of F{ with the edges in F2 removed.