On star-forest ascending subgraph decomposition

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph $G$ with ${n+1\choose 2}$ edges admits an edge decomposition $G=H_1\oplus\cdots \oplus H_n$ such that $H_i$ has $i$ edges and it is isomorphic to a subgraph of $H_{i+1}$, $i=1,\ldots ,n-1$. We show that every bipartite graph $G$ with ${n+1\choose 2}$ edges such that the degree sequence $d_1,\ldots ,d_k$ of one of the stable sets satisfies $ d_{k-i}\ge n-i\; \text{for each}\; 0\le i\le k-1,$, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.


Introduction
A graph G with n+1 2 edges has an Ascending Subgraph Decomposition (ASD) if it admits an edge-decomposition G = G 1 ⊕ · · · ⊕ G n such that G i has i edges and it is isomorphic to a subgraph of G i+1 , 1 ≤ i < n. Throughout this paper we use the symbol ⊕ to denote edge-disjoint union of graphs. It was conjectured by Alavi, Boals, Chartrand, Erdős and Oellerman [1] that every graph of size n+1 2 admits an ASD. The conjecture has been proved for a number of particular cases, including forests [5], regular graphs [9], complete multipartite graphs [8] or graphs with maximum degree ∆ ≤ n/2 [6].
In the same paper Alavi et al. [1] conjectured that every star forest of size n+1 2 in which each connected component has size at least n admits an ASD in which every graph in the decomposition is a star. This conjecture was proved by Ma, Zhou and Zhou [13], and the condition was later on weakened to the effect that the second smaller component of the star forest has size at least n by Chen, Fu , Wang and Zhou [4].
The class of bipartite graphs which admits a star-forest-ASD is clearly larger than the one which admit star -ASD, see Figure 1 for a simple example. This motivates the study of star-forest-ASD for bipartite graphs in terms of the degree sequence of one of the stable sets, which is the purpose of this paper.
Faudree, Gyárfás and Schelp [5] proved that every forest of stars admits a star-forest-ASD. These authors mention, in the same paper, that "Surprisingly [this result], is the most difficult to prove. This could indicate that the conjecture (if true) is a difficult one to prove". In the same paper the authors propose the following question: Let G be a graph = ⊕ ⊕ Figure 1. Bipartite graph with Star-Forest-ASD but with no Star-ASD.
with n+1 2 edges. Does G have an ASD such that each member is a star forest? In this paper we address this question for bipartite graphs when the centres of the stars in the star-forest-ASD belong to the same stable set. Our main result is the following one.
then, there is a star-forest-ASD of G.
The proof of Theorem 1 is made in two steps. First we prove the result for a class of bipartite graphs which we call reduced graphs. For this we use a representation of a star-forest decomposition by the so-called ascending matrices and certain multigraphs and reduce the problem to the existence of a particular edge-coloring of these multigraphs. The terminology and the proof for reduced graphs is contained in Section 2. In Section 3 we present an extension lemma, which uses a result of Häggkvist [10] on list edge-colorings, which allows one to extend the decomposition from reduced to all bipartite graphs with the same degree sequence on one of the stable sets, completing the proof of Theorem 1. The final section contains some concluding remarks.
The sufficient condition on the degrees given in Theorem 1 is not far from being necessary. Lemma 1. If every bipartite graph G with degree sequence d 1 ≤ d 2 ≤ · · · ≤ d k of one stable sets X of G admits a star-forest decomposition (ascending or not) with the centers of the stars in X then

Star-forest-ASD for reduced bipartite graphs
Thorughout the section G = G(X, Y ) denotes a bipartite graph with color classes X = {x 1 , . . . , x k } and Y = {y 1 , . . . , y n }. We denote by d X = (d 1 ≤ · · · ≤ d k ), d i = d(x i ), the degree sequence of the vertices in the stable set X of G. We call d X the X-degree sequence of G. We focus on star forest ASD with the stars of the decomposition centered at the vertices in X.
We first introduce some definitions. x 1 Remark 1. Let F, F ′ be two forests of stars with centers x 1 , . . . , x k and x ′ 1 , . . . , x ′ k respectively. Then F is isomorphic to a subgraph of Given two sequences d = (d 1 ≤ · · · ≤ d k ) and b = (b 1 ≤ · · · ≤ b n ) of nonnegative integers with i d i = j b j , denote by N (d, b) the set of k × n matrices A with nonnegative integer entries such that the row sums satisfy j a ij = d i , i = 1, . . . , k and the column sums satisfy i a ij = b j , j = 1, . . . , n.
Definition 2 (Ascending matrix). We say that a matrix A ∈ N (d, b) if in addition to the row sum being the sequence d and the column sum the sequence b, we have where A j denotes the j-th column of A.
For convenience we use the following notation for sequences. The constant sequence with r entries equal to x is denoted by x r and (x r , y s ) denotes the concatenation of x r and y s . Also, for an integer x we denote by x − the ascending sequence x − = (1, 2, . . . , x − 1, x). Sums and differences of sequences of the same length are understood to be componentwise.
We will use appropriate ascending matrices to define multigraphs which will lead to starforest-ASD as stated in Proposition 1 below. We recall that the bipartite adjacency matrix of a bipartite multigraph H with color classes We need a last definition which is borrowed from [10].
on the color class X has a sequential coloring for X if there is a proper edge coloring of H such that the edges incident with vertex x i receive colors {1, . . . , d i } for each i.
has a star-forest-ASD with centers of stars in X if and only if there is an A ∈ N (d X , n − ) such that the bipartite multigraph H = H(X, Z) with bipartite incidence matrix A admits a sequential coloring.
which admits a star-forest-ASD G = F 1 ⊕ · · · ⊕ F n with the centers of the stars in X.
Next we define an edge-coloring of H as follows. Denote by N F j (x i ) = {y i 1 , . . . , y is } the set of neighbours of x i in the forest F j . Then x i is joined in H to z j with s parallel edges.
We color these edges with the subscripts {i 1 , . . . , i s } of the neighbours of x i in F j , by assigning one of these colors to each parallel edge bijectively. In other words, if we define then the coloring is defined by any bijection from the |I ij | parallel edges joining x i with z j in H to I ij .
Since the original graph G is simple and the star-forests F 1 , . . . , F n form a decomposition of G, no two edges incident to a vertex x i receive the same color. On the other hand, since each star-forest F j has its stars centered in vertices in X, and therefore each vertex in Y has degree at most one in each F j , by the bijections which define the coloring, no two edges incident to z j receive the same color. Hence, the coloring is proper. Moreover, since the graph G is reduced, the edges incident to x i receive the colors {1, . . . , d i } and the coloring is sequential. This completes the if part of the proof.
Reciprocally, assume that A ∈ N (d X , n − ) and that the multigraph H = H(X, Z) with bipartite adjacency matrix A has a sequential coloring. Each z j will be associated to the subgraph F j of G defined as follows. For each edge x i z j of color h we declare the edge x i y h to be in F j . This way we obtain a subgraph of G because h ≤ d(x i ) (the coloring is sequential) and the graph G is reduced. Moreover, since the coloring is proper, the degree of every vertex y h in F j is at most one. Hence F j is a forest of stars and it has j edges. Moreover, for j = j ′ , the subgraphs F j and F j ′ are edge-disjoint again from the fact that the coloring is proper. Finally, since the matrix A is ascending, F i is isomorphic to a subgraph of F i+1 for each i = 1, · · · , n − 1. Hence, G = F 1 ⊕ · · · ⊕ F n is a starforest-ASD for G. This completes the proof. Figure 3 illustrates the statement of Proposition 1.
In order to prove the main result for reduced graphs we will show the existence of an appropriate ascending matrix and that the multigraph associated to it admits a sequential coloring.
According to Remark 1, the above mentioned result [5] on the existence of a star-forest-ASD of every star forest with n+1 2 edges can be reformulated as the existence of an ascending matrix A ∈ N (d, n − ) for every sequence d = ( x 2 x 3 x 4 x 5 Lemma 2. For every sequence d = (d 1 ≤ · · · ≤ d k ) there is a (d, n − )-ascending matrix C.
We next show that there exists an ascending matrix A ∈ N (d, n − ) of a particular shape that will be useful to prove the existence of star-forest-ASD for reduced graphs with that degree sequence.
We note that, if b 1 ≤ · · · ≤ b n , then each matrix A ∈ N (a, b) with (0, 1) entries is ascending. The support of a matrix B is the set of positions with nonzero entries. We observe that if B ∈ N (a, b), T ∈ N (a ′ , b ′ ) and B and T have disjoint support and the same dimensions, then B + T ∈ N (a + a ′ , b + b ′ ). The last sentence also holds if the support of T and B intersect in a square submatrix and T has constant entries in this submatrix. The above observations will be used in the proof of the next Lemma, which is illustrated in Figure 2. There is a matrix A ∈ N (d, n − ) such that a ij ≥ 1 for each (i, j) with i + j ≥ k + 1.
Proof. Consider the (k × n) matrix T with t ij = 1 for i + j ≥ k + 1 and t ij = 0 otherwise.
Extend this matrix to a k × n matrix A ′ by adding zero columns to the right. Since the last (n − k) columns of T are the all-ones vectors, the matrix A = A ′ + T still has the ascending column property and, by construction, it is in N (d, n − ) with nonzero entries in the positions (i, j) with i + j ≥ k + 1.   (4,6,9,9).
Next Lemma gives a sufficient condition for a degree sequence of a reduced graph to admit a star-forest-ASD.
then the reduced graph with degree sequence d admits a star-forest ASD.
Proof. Let A ∈ N (d, n − ) such that a ij ≥ 1 for each (i, j) with i + j ≥ k + 1, whose existence is ensured by Lemma 3.
Let H be the bipartite multigraph with stable sets X = {x 1 , . . . , x k } and Z = {z 1 , . . . , z n } whose bipartite adjacency matrix is A. We next show that H admits a sequential coloring. The result will follow by Proposition 1. Such matchings exist in H by the condition a ij ≥ 1 for each pair (i, j) with i + j ≤ k + 1, and they are pairwise edge-disjoint. For each j = 0, 1, . . . , k − 1, let M j ⊂ M ′ j be obtained by selecting from M ′ j the edges incident to x i whenever α i ≥ j. In this way each x i is incident with the matchings M 1 , . . . , M t(i) with t(i) = min{k, α i } and has degree at least n − k in M 1 ⊕ · · · ⊕ M k . On the other hand, by the condition on the degree sequence d X , since α i ≥ k − i + 1, the vertex z n−i is incident to k − i edges in M 1 ⊕ · · · ⊕ M k .
Let H ′ denote the bipartite multigraph obtained from H by removing the edges in  (4,6,9,9) depicted by their bipartite adjacency matrices.
Let ∆ ′ (X) be the maximum degree in H ′ of the vertices in X. If ∆ ′ (X) > n−k then there is a matching M ′ ∆ ′ (X) in H ′ from the vertices of maximum degree in X to Z. Color the edges of this matching with ∆ ′ (X). By removing this matching from H ′ we obtain a bipartite multigraph in which the maximum degree of vertices in X is ∆ ′ (X) − 1. By iterating this process we eventually reach a bipartite multigraph H ′′ in which every vertex in X has degree n−k while the maximum degree of the vertices in Z still satisfies ∆ ′′ (Z) ≤ n−k. By König's theorem, the edge-chromatic number of H ′′ is n − k. By combining an edge-coloring of H ′′ with n − k colors with the ones obtained in the process of reducing the maximum degree of H 1 , the multigraph H ′ can be properly edge-colored in such a way that vertex x i is incident in H ′ with colors {1, . . . , d ′ i }, 1 ≤ i ≤ k. We finally obtain a sequential coloring of H by adding to H ′ the matchings M 1 , . . . , M k , where M j gets color ∆ ′ (X) + (k − j): the largest color of an edge incident to x i is d H ′ (x i ) + t(i) = d i . By Proposition 1, G has a starforest-ASD.
The next result is a reformulation of Lemma 1 which shows that, if a reduced bipartite graph can be decomposed into n star forests with sizes 1, 2, . . . , n, regardless of the fact that it is ascending, then the degree sequence of the graph satisfies a condition which is necessary for the existence of a star-forest-ASD.
Lemma 5. Let G = (X ∪ Y, E) be a reduced bipartite graph with degree sequence (d 1 ≤ · · · ≤ d k ). If G admits an edge-decomposition into stars forests with centres in X, where F i has i edges, then (1) Proof. Let X = {x 1 , . . . , x k } and Y = {y 1 , . . . , y d k } be the bipartition of G where x i is adjacent to y 1 , . . . , y d i for each i. Since the graph is reduced, the last d k − d k−1 vertices of Y have degree one, the preceding d k−1 − d k−2 have degree 2 and, in general, the consecutive d j − d j−1 vertices y d j−1 +1 , . . . , y d j have degree j − 1.
Since F n has n leaves in Y we clearly have |Y | = d k ≥ n. Thus (1) is satisfied for t = 1.
Assume that (1) is satisfied for some t = j − 1 < k. Consider the subgraph G j of G induced by F n ⊕ F n−1 ⊕ · · · ⊕ F n−(j−1) . Since each vertex in Y has degree at most one in each forest, it has degree at most j in G j . By combining this remark with the former upper bound on the degrees of vertices in Y we have and (1) is satisfied for t = j. This concludes the proof.

An extension Lemma
In this section we prove an extension Lemma which shows that, if G R admits a star-forest-ASD then so does G. This reduces the problem of giving sufficient conditions on the degree sequence of one stable set to ensure the existence of a star-forest-ASD to bipartite reduced graphs. For the proof of our extension Lemma we use the following result by Häggkvist [10] on edge list-colorings of bipartite multigraphs.
where each F ′ i is a star forest, then G has an edge decomposition Proof. Let C be the (k × t) matrix whose entry c ij is the number of edges incident to a i in the star forest F ′ j of the edge decomposition of G R . Consider the bipartite multigraph H with stable sets A and U = {u 1 , . . . , u t }, where a i is joined with u j with c ij parallel edges. Now, for each pair (i, j), color the c ij parallel edges of H with the neighbors of a i in the forest F ′ j bijectively. Note that in this way we get a proper edge-coloring of H: two edges incident with a vertex a i receive different colors since the bipartite graph G R has no multiple edges, and two edges incident to a vertex u j receive different colors since F ′ j is a star forest. By the definition of the bipartite graph G R , each vertex a i ∈ A is incident in the bipartite multigraph H with edges colored 1, 2, . . . , d i . Let L(a i ) be the list of neighbours of a i in the original bipartite graph G. By Theorem 2, there is a proper edge-coloring χ ′ of H in which the edges incident to vertex a i in A receive the colors from the list L(a i ) for each i = 1, . . . , k. Now construct F s by letting the edge a i b j be in F s whenever the edge a i u s is colored b j in the latter edge-coloring of H. Thus F s has the same number of edges than F ′ s and the degree of a i in F s is c is , the same as in F ′ s . Moreover, since the coloring is proper, F ′ s is a star forest and two forests F ′ s , F ′ s ′ are edge-disjoint whenever s = s ′ . This concludes the proof.
We are now ready to prove our main result.
Proof of Theorem 1: Let G be a bipartite graph with degree sequence d = (d 1 ≥ · · · ≥ d k ) satisfying i d i = n+1 2 and d i ≥ n − i+ 1, i = 1, . . . , k. By Lemma 2 there is a d-ascending matrix C such that c ij ≥ 1 for each (i, j) with i + j ≤ k + 1. By Lemma 4 there is a starforest-ASD of the reduced graph G R with degree sequence d. By Lemma 6 there is also a star-forest-ASD for G.