- Email: [email protected]
Decomposing a graph into bistars
Journal of Combinatorial Theory, Series B 103 (2013) 504–508
Contents lists available at SciVerse ScienceDirect
Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb
Decomposing a graph into bistars Carsten Thomassen a,b a b
Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark King Abdulaziz University, Jeddah, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 20 December 2010 Available online 3 June 2013 Keywords: Orientations modulo k Bistar decomposition 3-Flow conjecture
a b s t r a c t Bárat and the present author conjectured that, for each tree T , there exists a natural number k T such that the following holds: If G is a k T -edge-connected graph such that | E ( T )| divides | E (G )|, then G has a T -decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S (k, k + 1). © 2013 Elsevier Inc. All rights reserved.
1. Introduction In [11] it was proved that a graph of large (but fixed) edge-connectivity has an orientation of the edges such that each vertex has prescribed outdegree modulo k provided the obvious necessary condition holds. More precisely: Theorem 1. Let k be any natural number. If G is a (2k2 + k)-edge-connected graph, and, for each vertex x, p (x) is a natural number such that the size (number of edges) of G is congruent to the sum of all p (x) modulo k, then the edges of G can be oriented such that the outdegree of each vertex x is congruent to p (x) modulo k. For k = 3 the edge-connectivity 8 suffices. The case k = 3 proves a weakening of Tutte’s 3-flow conjecture called the weak 3-flow conjecture proposed by Jaeger [5]. For general k, Theorem 1 proves the analogous weakening of the circular flow conjecture proposed by Jaeger in [4]. As an immediate consequence of Theorem 1, the conjecture by Bárat and Thomassen [2] now holds for each star. The present paper is a continuation of [11]. Let S (k) denote the star with k edges. Let S (k, p ) denote the tree with diameter 3 and two adjacent vertices of degrees k, p, respectively. We shall here
E-mail address: [email protected]. 0095-8956/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jctb.2013.05.003
C. Thomassen / Journal of Combinatorial Theory, Series B 103 (2013) 504–508
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apply Theorem 1 to verify the decomposition conjecture for each bistar S (k, k + 1). These are the first trees which satisfy the decomposition conjecture and which are neither stars nor paths. Combined with earlier results [9,10,12,11], the conjecture has now been verified for all trees with at most 4 edges. However, the methods used for these small trees are very much different (except for the stars), and also the conjecture is open for each tree with 5 edges, except the star S (5). So, the decomposition conjecture is still wide open. For example, it seems that a new method is needed even for the path with 5 edges. There are two obvious ideas one may use to attack the decomposition conjecture. The first is edgeliftings (for the definition, see e.g. [7]). If T is a tree for which the decomposition conjecture holds, then one would expect it to hold for the subdivision graph (that is, the tree obtained by inserting a vertex on each edge.) And one would expect that it can be proved using liftings. The main problem with liftings is that they may create multiple edges, and the decomposition conjecture is not valid for multigraphs. This caused much additional work in [9]. In [12] we got around this problem by working in a more general framework which, however, seems to work for paths only. We also use liftings in the present paper. This is possible because, after the liftings, we use the orientation results in [11] which are valid for multigraphs. The other obvious idea is the following: Suppose G is a graph which we wish to decompose into copies of the tree T , say. Then write T = T 1 ∪ T 2 , G = G 1 ∪ G 2 , decompose G i into copies of T i , i = 1, 2, and finally form a matching of the copies of T 1 with the copies of T 2 . I was unable to do this in the smallest obvious examples, namely the path or star with 4 edges. In order to make this idea work it seems that one needs some more information on the local structure of the decomposition, and it is not clear how to formulate the relevant condition. In this paper we use this idea for the bistars S (k, k + 1), because these trees are the only trees with the following property: If T 1 , T 2 are two intersecting edge-disjoint trees in a bipartite graph, each isomorphic to the star S (k), and with centers in opposite sides, then T 1 ∪ T 2 = S (k, k + 1). It is not obvious how to make the above decomposition G = G 1 ∪ G 2 , though. For that we use Theorem 1. 2. Reducing the decomposition conjecture to bipartite graphs In the last remark of the previous section it is important that the graph in which we form the union T 1 ∪ T 2 = S (k, k + 1) is bipartite. We shall therefore reduce the decomposition conjecture to the bipartite case. This reduction was done in [9,10,12] when the tree is a path. The general reduction is this section was done independently by Bárat and Gerbner [1]. If T is a tree, then we define, if possible, k T as the smallest natural number satisfying the following: If G is a k T -edge-connected graph such that | E ( T )| divides | E (G )|, then G has a T -decomposition. We also define, if possible, kT as the smallest natural number satisfying the following: If G is a k T -edge-connected bipartite graph such that | E ( T )| divides | E (G )|, then G has a T -decomposition. In this section we prove that the existence of kT implies the existence of k T . Recall that if v is a vertex in a graph G, then the degree, also called the G-degree, of v is denoted d( v , G ) (or just d( v )). By a rooted forest we mean a forest where each component has a special vertex, called a root. Proposition 1. Let d, p, q be natural numbers such that d 2. Let G be a graph, and let T 1 , T 2 , . . . , T m be a collection of rooted forests each of maximum degree at most d such that each vertex has distance at most q to the root. For each root r i , j of the forests T i , let v i , j be a vertex of G. Let F i be a set of vertices in G such that | V ( T i )| + | F i | − 1 p for i = 1, 2, . . . , m. (We think of the graphs G , T 1 , T 2 , . . . , T m as being pairwise disjoint. The vertices v i ,1 , v i ,2 , . . . are distinct, but v i , j , v i , j need not be distinct if i = i .) If v is a vertex such that there is no i , j such that v i , j = v, then we put f ( v ) = 1. Otherwise, let f ( v ) be the number of pairs i , j such that v i , j = v. Assume that, for each vertex v in G,
d( v , G ) 2 f ( v ) · (dpq)2q .
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of pairwise edge-disjoint rooted subforests such that T is Then G contains a collection T 1 , T 2 , . . . , T m i isomorphic to T i with the root r i , j corresponding to the root v i , j in T i (for each j = 1, 2, . . .) and contains no vertex of F i for i = 1, 2, . . . , m.
Proof. The proof is by induction on q. The edges of G can be directed such that the outdegree and indegree of any vertex differ by at most 1. If q = 1 we can use all edges leaving v to represent the trees rooted at v. For later purposes we observe that for this we only need the inequality
d( v , G ) 2 df ( v ) + p . The reason for the number p in this inequality is that we must avoid the | F i | vertices in F i . But, we must also avoid the vertices in the other components in T i . There are at most | V ( T i )| − 1 other vertices, and p is chosen to big enough for this. So assume that q 2. For each vertex v in G we split v into two vertices u 1 , u 2 such that all edges leaving (respectively entering) v leave u 2 (respectively enter u 1 ). Then we further split u 1 (respectively u 2 ) such that each vertex except possibly one has indegree (respectively outdegree) precisely (dpq)2q . As the resulting digraph has an underlying undirected bipartite graph of maximum degree (dpq)2q , it has a proper edge-coloring with colors 1, 2, . . . , (dpq)2q . Now consider a vertex v in G. The vertex u 2 is split up into at least f ( v ) vertices of outdegree precisely (dpq)2q . Select precisely f ( v ) of these vertices. For each of these we use the edges of colors 1, 2, . . . , d + p to realize all the edges incident with all the roots in those forests T i for which v is a root. Those edges will be part of the forests T i . We now delete those edges both from G and also from each T i , and we add v to F i whenever v is a root of T i . Then we use induction. This is possible because the new f ( v ) is at most
f ( v ) = (d + p ) d( v , G )/ 2(dpq)2q
.
The new degree of v is greater than
d ( v ) = d( v , G ) − df ( v ) − f ( v ). So, in order to use induction it suffices to verify that
2q−2
d ( v ) 2 f ( v ) · dp (q − 1)
.
It suffices to prove this inequality when
d( v , G ) = 2 f ( v ) · (dpq)2q . (For, if we keep d, p , q, f ( v ) fixed, then the left-hand side grows faster than the right-hand side when we think of these as functions of d( v , G ).) We leave this to the reader. This proves Proposition 1. 2 Theorem 2. Let T be a tree on n vertices such that kT exists. Then also k T exists, and
k T 4kT + 16(n − 1)6n−5 . If, in addition, T has diameter at most 3, then
k T 4kT + 16n(n − 1). Proof. Let G be any graph of edge-connectivity at least 4kT + 16(n − 1)6n−5 whose size is divisible by n − 1. By Proposition 1 in [12], G has a spanning bipartite graph H of edge-connectivity at least 2kT + 8(n − 1)6n−5 . By [3,8,13], H has at least kT + 4(n − 1)6n−5 pairwise edge-disjoint spanning trees.
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Divide H into three graphs H 1 , H 2 , H 3 such that H 1 has at least 2(n − 1)6n−5 pairwise edge-disjoint spanning trees, H 2 has at least 2(n − 1)6n−5 pairwise edge-disjoint spanning trees, and H 3 has at least kT pairwise edge-disjoint spanning trees. Let A 1 , A 2 be the partite classes of H . We claim that, for i = 1, 2, G ( A i ) ∪ H i has a collection of edge-disjoint copies of T such that each edge of G ( A i ) is contained in one of those trees. This will complete the proof because we then apply the existence of kT to those edges that are not contained in any copy of T . Those edges form a bipartite graph containing H 3 , and therefore the edge-connectivity of the graph consisting of the uncovered edges is at least kT . To prove the claim we focus on G ( A 1 ) ∪ H 1 . (G ( A 2 ) ∪ H 2 is treated analogously.) We first delete the edges of as many pairwise edge-disjoint copies of T in G ( A 1 ) as possible. The resulting subgraph of G ( A 1 ) (which we also call G ( A 1 )) has no tree isomorphic to T . We call an endvertex of T the root of T . Now we construct (and delete the edges of) a sequence of trees T 1 , T 2 . . . in G ( A 1 ) each of which is a rooted subtree of T . We begin the construction of each T i by letting a vertex of maximum degree in the current G ( A 1 ) be the root. Then we extend that root to a maximal tree in the current G ( A 1 ) which is isomorphic to a subtree T i of T . If v is a vertex in G ( A 1 ) which is also a vertex in T i such that the degree of v in T i is strictly smaller that the corresponding degree in T , then we say that v is hungry in T i . Let us assume that v is a vertex which is hungry in T i but not in any T j with j < i.
The only edges incident with v and not used by T i or by a previous tree are edges from v to T i . There are at most n − 2 such edges. This means that v may be hungry in at most n − 2 subsequent trees, that is, a vertex in G ( A 1 ) is hungry in at most n − 1 trees. Now we use Proposition 1 (with each F i being empty) to extend in H 1 the subtrees to trees isomorphic to T . We put f ( v ) = d = p = q = n − 1. If T has diameter 3, then we argue slightly differently. When we construct the collection of maximal subtrees of T in G ( A 1 ) we begin by selecting an edge in G ( A 1 ) which represent the middle edge of T . Apart from that we repeat the argument above. When we use Proposition 1 we put f ( v ) = n − 1, and d = p = n − 1, as above. But now it suffices to put q = 1, and in this case we only need the inequality
d( v , G ) 2 df ( v ) + p . This proves Theorem 2.
2
3. Bistar decomposition of bipartite graphs Proposition 2. Let k be a natural number. If G is a (4k2 + 6k)-edge-connected, bipartite graph with partite classes A 1 , A 2 such that the size of G is divisible by k, then G can be divided into two spanning edge-disjoint, k-edge-connected graphs G 1 , G 2 such that all vertices of A 1 have G 1 -degrees divisible by k, and all vertices of A 2 have G 2 -degrees divisible by k. Proof. By [3,8,13], G has 2k2 + 3k pairwise edge-disjoint spanning trees. Hence we can divide G into three spanning edge-disjoint graphs H 1 , H 2 , H 3 such that H i has k pairwise edge-disjoint spanning trees for i = 1, 2, and H 3 has 2k2 + k pairwise edge-disjoint spanning trees. Now we apply Theorem 1 to H 3 . If x is a vertex in A 1 , we put p (x) = k − d(x, H 1 ). If x is a vertex in A 2 , we put p (x) = k − d(x, H 2 ). After the orientation of the edges in H 3 we add to H 1 all edges in H 3 directed from A 1 to A 2 , and we call the resulting graph G 1 . Also, we add to H 2 all edges in H 3 directed from A 2 to A 1 , and we call the resulting graph G 2 . This proves Proposition 2. 2 Theorem 3. Let k be a natural number. If G is a 180k4 -edge-connected, bipartite graph with partite classes A 1 , A 2 such that the size of G is divisible by 2k, then G can be divided into pairwise edge-disjoint copies of the bistar S (k, k + 1). Proof. We let G 1 , G 2 be as in Proposition 2 with 2k2 + 2k instead of k. For this we need the edgeconnectivity of G to be at least 16k2 (k + 1)2 + 12k2 + 12k which is smaller than 88k4 . We also need
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the size of G to be divisible by 2k2 + 2k. In order to achieve this we first take 90k4 pairwise edgedisjoint spanning trees in G. We use 88k4 of these to ensure the edge-connectivity. In the remaining 2k4 trees we successively delete pairwise edge-disjoint copies of S (k, k + 1) until the remaining graph has size divisible by 2k2 + 2k. (For this we need to delete at most k copies of S (k, k + 1), and this is clearly possible in a graph of minimum degree at least 2k2 because the minimum degree of the graph decreases by at most k + 1 when you delete a copy of S (k, k + 1).) In G 1 every vertex in A 1 has even degree. By Mader’s lifting theorem [7] (or a special case by Lovász [6]), all edges incident with such a vertex in A 1 can be lifted such that we preserve the edge-connectivity. We do this for all vertices in A 1 and obtain a graph G 1 on the vertex set A 2 and of edge-connectivity at least 2k2 + 2k. The size of G 1 is divisible by 2k, and hence the size of G 1 is divisible by k. By Theorem 1, the edges of G 1 can be directed such that each outdegree is divisible by k. Consider an edge of G 1 . That edge corresponds to a directed path of length 2 in G 1 . We color the first of these edges red and the last edge blue. The last blue edge is called the successor of the red edge. Clearly, the red edges leaving a vertex in A 2 can be divided into red stars S (k) which we call red stars of type 1 (because their edges are from G 1 ). As a vertex in A 1 has G 1 -degree divisible by 2k, the blue edges leaving that vertex can be divided into blue stars S (k) which we call blue stars of type 1. We now form a new bipartite graph M as follows. A vertex in M is one of the stars defined above. A vertex in M corresponding to a red star is joined precisely to those k blue stars which contain successors of the red edges in the red star. As M is bipartite and k-regular, M has a perfect matching. Any edge in this matching corresponds to a red star and a blue star. These two stars intersect, and therefore, their union is a bistar S (k, k + 1). Hence G 1 can be decomposed into bistars. Similarly, G 2 can be decomposed into bistars. This proves Theorem 3. 2 It is well known and easy to see that every connected graph with an even number of edges can be divided into pairwise edge-disjoint copies of the bistar S (1, 2). Theorem 2 and Theorem 3 now prove the following for all bistars of the form S (k, k + 1). Theorem 4. Let k be a natural number. If G is a 784k4 -edge-connected graph such that the size of G is divisible by 2k, then G can be divided into pairwise edge-disjoint copies of the bistar S (k, k + 1). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J. Bárat, D. Gerbner, Edge-decomposition of graphs into copies of a tree with four edges, manuscript. J. Bárat, C. Thomassen, Claw-decompositions and Tutte-orientations, J. Graph Theory 52 (2006) 135–146. J. Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Natl. Bur. Stand. Sect. B 69 (1965) 67–72. F. Jaeger, On circular flows in graphs, Proc. Colloq. Math. János Bolyai (1982) 391–402. F. Jaeger, Nowhere-zero flow problems, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory 3, Academic Press, 1988, pp. 71–95. L. Lovász, On some connectivity properties of Eulerian graphs, Acta Math. Acad. Sci. Hung. 28 (1976) 129–138. W. Mader, A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978) 145–164. C.St.J.A. Nash-Williams, Edge-disjoint spanning trees of finite graphs, J. London Math. Soc. 36 (1961) 445–450. C. Thomassen, Decompositions of highly connected graphs into paths of length 4, Abh. Math. Sem. Univ. Hamburg 78 (2008) 17–26. C. Thomassen, Decompositions of highly connected graphs into paths of length 3, J. Graph Theory 58 (2008) 286–292. C. Thomassen, The weak 3-flow conjecture, J. Combin. Theory Ser. B 102 (2012) 521–529. C. Thomassen, Decomposing graphs into paths of fixed length, Combinatorica 33 (2013) 97–123. W.T. Tutte, On the problem of decomposing a graph into n connected factors, J. London Math. Soc. 36 (1961) 221–230.
Contents lists available at SciVerse ScienceDirect
Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb
Decomposing a graph into bistars Carsten Thomassen a,b a b
Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark King Abdulaziz University, Jeddah, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 20 December 2010 Available online 3 June 2013 Keywords: Orientations modulo k Bistar decomposition 3-Flow conjecture
a b s t r a c t Bárat and the present author conjectured that, for each tree T , there exists a natural number k T such that the following holds: If G is a k T -edge-connected graph such that | E ( T )| divides | E (G )|, then G has a T -decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S (k, k + 1). © 2013 Elsevier Inc. All rights reserved.
1. Introduction In [11] it was proved that a graph of large (but fixed) edge-connectivity has an orientation of the edges such that each vertex has prescribed outdegree modulo k provided the obvious necessary condition holds. More precisely: Theorem 1. Let k be any natural number. If G is a (2k2 + k)-edge-connected graph, and, for each vertex x, p (x) is a natural number such that the size (number of edges) of G is congruent to the sum of all p (x) modulo k, then the edges of G can be oriented such that the outdegree of each vertex x is congruent to p (x) modulo k. For k = 3 the edge-connectivity 8 suffices. The case k = 3 proves a weakening of Tutte’s 3-flow conjecture called the weak 3-flow conjecture proposed by Jaeger [5]. For general k, Theorem 1 proves the analogous weakening of the circular flow conjecture proposed by Jaeger in [4]. As an immediate consequence of Theorem 1, the conjecture by Bárat and Thomassen [2] now holds for each star. The present paper is a continuation of [11]. Let S (k) denote the star with k edges. Let S (k, p ) denote the tree with diameter 3 and two adjacent vertices of degrees k, p, respectively. We shall here
E-mail address: [email protected]. 0095-8956/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jctb.2013.05.003
C. Thomassen / Journal of Combinatorial Theory, Series B 103 (2013) 504–508
505
apply Theorem 1 to verify the decomposition conjecture for each bistar S (k, k + 1). These are the first trees which satisfy the decomposition conjecture and which are neither stars nor paths. Combined with earlier results [9,10,12,11], the conjecture has now been verified for all trees with at most 4 edges. However, the methods used for these small trees are very much different (except for the stars), and also the conjecture is open for each tree with 5 edges, except the star S (5). So, the decomposition conjecture is still wide open. For example, it seems that a new method is needed even for the path with 5 edges. There are two obvious ideas one may use to attack the decomposition conjecture. The first is edgeliftings (for the definition, see e.g. [7]). If T is a tree for which the decomposition conjecture holds, then one would expect it to hold for the subdivision graph (that is, the tree obtained by inserting a vertex on each edge.) And one would expect that it can be proved using liftings. The main problem with liftings is that they may create multiple edges, and the decomposition conjecture is not valid for multigraphs. This caused much additional work in [9]. In [12] we got around this problem by working in a more general framework which, however, seems to work for paths only. We also use liftings in the present paper. This is possible because, after the liftings, we use the orientation results in [11] which are valid for multigraphs. The other obvious idea is the following: Suppose G is a graph which we wish to decompose into copies of the tree T , say. Then write T = T 1 ∪ T 2 , G = G 1 ∪ G 2 , decompose G i into copies of T i , i = 1, 2, and finally form a matching of the copies of T 1 with the copies of T 2 . I was unable to do this in the smallest obvious examples, namely the path or star with 4 edges. In order to make this idea work it seems that one needs some more information on the local structure of the decomposition, and it is not clear how to formulate the relevant condition. In this paper we use this idea for the bistars S (k, k + 1), because these trees are the only trees with the following property: If T 1 , T 2 are two intersecting edge-disjoint trees in a bipartite graph, each isomorphic to the star S (k), and with centers in opposite sides, then T 1 ∪ T 2 = S (k, k + 1). It is not obvious how to make the above decomposition G = G 1 ∪ G 2 , though. For that we use Theorem 1. 2. Reducing the decomposition conjecture to bipartite graphs In the last remark of the previous section it is important that the graph in which we form the union T 1 ∪ T 2 = S (k, k + 1) is bipartite. We shall therefore reduce the decomposition conjecture to the bipartite case. This reduction was done in [9,10,12] when the tree is a path. The general reduction is this section was done independently by Bárat and Gerbner [1]. If T is a tree, then we define, if possible, k T as the smallest natural number satisfying the following: If G is a k T -edge-connected graph such that | E ( T )| divides | E (G )|, then G has a T -decomposition. We also define, if possible, kT as the smallest natural number satisfying the following: If G is a k T -edge-connected bipartite graph such that | E ( T )| divides | E (G )|, then G has a T -decomposition. In this section we prove that the existence of kT implies the existence of k T . Recall that if v is a vertex in a graph G, then the degree, also called the G-degree, of v is denoted d( v , G ) (or just d( v )). By a rooted forest we mean a forest where each component has a special vertex, called a root. Proposition 1. Let d, p, q be natural numbers such that d 2. Let G be a graph, and let T 1 , T 2 , . . . , T m be a collection of rooted forests each of maximum degree at most d such that each vertex has distance at most q to the root. For each root r i , j of the forests T i , let v i , j be a vertex of G. Let F i be a set of vertices in G such that | V ( T i )| + | F i | − 1 p for i = 1, 2, . . . , m. (We think of the graphs G , T 1 , T 2 , . . . , T m as being pairwise disjoint. The vertices v i ,1 , v i ,2 , . . . are distinct, but v i , j , v i , j need not be distinct if i = i .) If v is a vertex such that there is no i , j such that v i , j = v, then we put f ( v ) = 1. Otherwise, let f ( v ) be the number of pairs i , j such that v i , j = v. Assume that, for each vertex v in G,
d( v , G ) 2 f ( v ) · (dpq)2q .
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of pairwise edge-disjoint rooted subforests such that T is Then G contains a collection T 1 , T 2 , . . . , T m i isomorphic to T i with the root r i , j corresponding to the root v i , j in T i (for each j = 1, 2, . . .) and contains no vertex of F i for i = 1, 2, . . . , m.
Proof. The proof is by induction on q. The edges of G can be directed such that the outdegree and indegree of any vertex differ by at most 1. If q = 1 we can use all edges leaving v to represent the trees rooted at v. For later purposes we observe that for this we only need the inequality
d( v , G ) 2 df ( v ) + p . The reason for the number p in this inequality is that we must avoid the | F i | vertices in F i . But, we must also avoid the vertices in the other components in T i . There are at most | V ( T i )| − 1 other vertices, and p is chosen to big enough for this. So assume that q 2. For each vertex v in G we split v into two vertices u 1 , u 2 such that all edges leaving (respectively entering) v leave u 2 (respectively enter u 1 ). Then we further split u 1 (respectively u 2 ) such that each vertex except possibly one has indegree (respectively outdegree) precisely (dpq)2q . As the resulting digraph has an underlying undirected bipartite graph of maximum degree (dpq)2q , it has a proper edge-coloring with colors 1, 2, . . . , (dpq)2q . Now consider a vertex v in G. The vertex u 2 is split up into at least f ( v ) vertices of outdegree precisely (dpq)2q . Select precisely f ( v ) of these vertices. For each of these we use the edges of colors 1, 2, . . . , d + p to realize all the edges incident with all the roots in those forests T i for which v is a root. Those edges will be part of the forests T i . We now delete those edges both from G and also from each T i , and we add v to F i whenever v is a root of T i . Then we use induction. This is possible because the new f ( v ) is at most
f ( v ) = (d + p ) d( v , G )/ 2(dpq)2q
.
The new degree of v is greater than
d ( v ) = d( v , G ) − df ( v ) − f ( v ). So, in order to use induction it suffices to verify that
2q−2
d ( v ) 2 f ( v ) · dp (q − 1)
.
It suffices to prove this inequality when
d( v , G ) = 2 f ( v ) · (dpq)2q . (For, if we keep d, p , q, f ( v ) fixed, then the left-hand side grows faster than the right-hand side when we think of these as functions of d( v , G ).) We leave this to the reader. This proves Proposition 1. 2 Theorem 2. Let T be a tree on n vertices such that kT exists. Then also k T exists, and
k T 4kT + 16(n − 1)6n−5 . If, in addition, T has diameter at most 3, then
k T 4kT + 16n(n − 1). Proof. Let G be any graph of edge-connectivity at least 4kT + 16(n − 1)6n−5 whose size is divisible by n − 1. By Proposition 1 in [12], G has a spanning bipartite graph H of edge-connectivity at least 2kT + 8(n − 1)6n−5 . By [3,8,13], H has at least kT + 4(n − 1)6n−5 pairwise edge-disjoint spanning trees.
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Divide H into three graphs H 1 , H 2 , H 3 such that H 1 has at least 2(n − 1)6n−5 pairwise edge-disjoint spanning trees, H 2 has at least 2(n − 1)6n−5 pairwise edge-disjoint spanning trees, and H 3 has at least kT pairwise edge-disjoint spanning trees. Let A 1 , A 2 be the partite classes of H . We claim that, for i = 1, 2, G ( A i ) ∪ H i has a collection of edge-disjoint copies of T such that each edge of G ( A i ) is contained in one of those trees. This will complete the proof because we then apply the existence of kT to those edges that are not contained in any copy of T . Those edges form a bipartite graph containing H 3 , and therefore the edge-connectivity of the graph consisting of the uncovered edges is at least kT . To prove the claim we focus on G ( A 1 ) ∪ H 1 . (G ( A 2 ) ∪ H 2 is treated analogously.) We first delete the edges of as many pairwise edge-disjoint copies of T in G ( A 1 ) as possible. The resulting subgraph of G ( A 1 ) (which we also call G ( A 1 )) has no tree isomorphic to T . We call an endvertex of T the root of T . Now we construct (and delete the edges of) a sequence of trees T 1 , T 2 . . . in G ( A 1 ) each of which is a rooted subtree of T . We begin the construction of each T i by letting a vertex of maximum degree in the current G ( A 1 ) be the root. Then we extend that root to a maximal tree in the current G ( A 1 ) which is isomorphic to a subtree T i of T . If v is a vertex in G ( A 1 ) which is also a vertex in T i such that the degree of v in T i is strictly smaller that the corresponding degree in T , then we say that v is hungry in T i . Let us assume that v is a vertex which is hungry in T i but not in any T j with j < i.
The only edges incident with v and not used by T i or by a previous tree are edges from v to T i . There are at most n − 2 such edges. This means that v may be hungry in at most n − 2 subsequent trees, that is, a vertex in G ( A 1 ) is hungry in at most n − 1 trees. Now we use Proposition 1 (with each F i being empty) to extend in H 1 the subtrees to trees isomorphic to T . We put f ( v ) = d = p = q = n − 1. If T has diameter 3, then we argue slightly differently. When we construct the collection of maximal subtrees of T in G ( A 1 ) we begin by selecting an edge in G ( A 1 ) which represent the middle edge of T . Apart from that we repeat the argument above. When we use Proposition 1 we put f ( v ) = n − 1, and d = p = n − 1, as above. But now it suffices to put q = 1, and in this case we only need the inequality
d( v , G ) 2 df ( v ) + p . This proves Theorem 2.
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3. Bistar decomposition of bipartite graphs Proposition 2. Let k be a natural number. If G is a (4k2 + 6k)-edge-connected, bipartite graph with partite classes A 1 , A 2 such that the size of G is divisible by k, then G can be divided into two spanning edge-disjoint, k-edge-connected graphs G 1 , G 2 such that all vertices of A 1 have G 1 -degrees divisible by k, and all vertices of A 2 have G 2 -degrees divisible by k. Proof. By [3,8,13], G has 2k2 + 3k pairwise edge-disjoint spanning trees. Hence we can divide G into three spanning edge-disjoint graphs H 1 , H 2 , H 3 such that H i has k pairwise edge-disjoint spanning trees for i = 1, 2, and H 3 has 2k2 + k pairwise edge-disjoint spanning trees. Now we apply Theorem 1 to H 3 . If x is a vertex in A 1 , we put p (x) = k − d(x, H 1 ). If x is a vertex in A 2 , we put p (x) = k − d(x, H 2 ). After the orientation of the edges in H 3 we add to H 1 all edges in H 3 directed from A 1 to A 2 , and we call the resulting graph G 1 . Also, we add to H 2 all edges in H 3 directed from A 2 to A 1 , and we call the resulting graph G 2 . This proves Proposition 2. 2 Theorem 3. Let k be a natural number. If G is a 180k4 -edge-connected, bipartite graph with partite classes A 1 , A 2 such that the size of G is divisible by 2k, then G can be divided into pairwise edge-disjoint copies of the bistar S (k, k + 1). Proof. We let G 1 , G 2 be as in Proposition 2 with 2k2 + 2k instead of k. For this we need the edgeconnectivity of G to be at least 16k2 (k + 1)2 + 12k2 + 12k which is smaller than 88k4 . We also need
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the size of G to be divisible by 2k2 + 2k. In order to achieve this we first take 90k4 pairwise edgedisjoint spanning trees in G. We use 88k4 of these to ensure the edge-connectivity. In the remaining 2k4 trees we successively delete pairwise edge-disjoint copies of S (k, k + 1) until the remaining graph has size divisible by 2k2 + 2k. (For this we need to delete at most k copies of S (k, k + 1), and this is clearly possible in a graph of minimum degree at least 2k2 because the minimum degree of the graph decreases by at most k + 1 when you delete a copy of S (k, k + 1).) In G 1 every vertex in A 1 has even degree. By Mader’s lifting theorem [7] (or a special case by Lovász [6]), all edges incident with such a vertex in A 1 can be lifted such that we preserve the edge-connectivity. We do this for all vertices in A 1 and obtain a graph G 1 on the vertex set A 2 and of edge-connectivity at least 2k2 + 2k. The size of G 1 is divisible by 2k, and hence the size of G 1 is divisible by k. By Theorem 1, the edges of G 1 can be directed such that each outdegree is divisible by k. Consider an edge of G 1 . That edge corresponds to a directed path of length 2 in G 1 . We color the first of these edges red and the last edge blue. The last blue edge is called the successor of the red edge. Clearly, the red edges leaving a vertex in A 2 can be divided into red stars S (k) which we call red stars of type 1 (because their edges are from G 1 ). As a vertex in A 1 has G 1 -degree divisible by 2k, the blue edges leaving that vertex can be divided into blue stars S (k) which we call blue stars of type 1. We now form a new bipartite graph M as follows. A vertex in M is one of the stars defined above. A vertex in M corresponding to a red star is joined precisely to those k blue stars which contain successors of the red edges in the red star. As M is bipartite and k-regular, M has a perfect matching. Any edge in this matching corresponds to a red star and a blue star. These two stars intersect, and therefore, their union is a bistar S (k, k + 1). Hence G 1 can be decomposed into bistars. Similarly, G 2 can be decomposed into bistars. This proves Theorem 3. 2 It is well known and easy to see that every connected graph with an even number of edges can be divided into pairwise edge-disjoint copies of the bistar S (1, 2). Theorem 2 and Theorem 3 now prove the following for all bistars of the form S (k, k + 1). Theorem 4. Let k be a natural number. If G is a 784k4 -edge-connected graph such that the size of G is divisible by 2k, then G can be divided into pairwise edge-disjoint copies of the bistar S (k, k + 1). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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