Edge-disjoint Subgraphs Research Articles

Simultaneous well-balanced orientations of graphs

Nash-Williams' well-balanced orientation theorem [C.St.J.A. Nash-Williams, On orientations, connectivity and odd-vertex-pairings in finite graphs, Canad. J. Math. 12 (1960) 555–567] is extended for orienting several graphs simultaneously. We prove that if G 1 , … , G k are pairwise edge-disjoint subgraphs of a graph G, then G has a well-balanced orientation G → such that the inherited orientations G → i of G i are well-balanced for all 1 ⩽ i ⩽ k . We also have new results about simultaneous well-balanced orientations of non-disjoint subgraphs of an Eulerian graph as well as those of different contractions of a graph.

Decompositions of infinite graphs: I—bond-faithful decompositions

We show that a graph can always be decomposed into edge-disjoint subgraphs of countable cardinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinal. We also show that this result, with the assumption of the Generalized Continuum Hypothesis, can be generalized to any uncountable cardinal. As applications of such decompositions we prove some results about Seymour's double-cover conjecture for infinite graphs, and about the maximal number of edge-disjoint spanning trees in graphs having high edge-connectivity. However, the main motivation for introducing these decompositions can be found in the second part of this paper where, to achieve a complete solution of the circuit decomposition problem (i.e. the problem of characterizing the graphs that admit decompositions into 2-regular connected subgraphs), we use the results of this first part to carry out a reduction to the countable case.

The Countable Character of Uncountable Graphs

We show that a graph can always be decomposed into edge-disjoint subgraphs of countable cardinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinals. We also show that the vertex set of any graph can be endowed with a well-ordering which has a certain compactness property with respect to edge-separation.

Packing of Kv with certain graphs of five vertices

Let K v be the complete graph on v vertices, and G a finite simple undirected graph without isolated vertices. A G-packing of K v , denoted by ( v, G,1)-packing, is a pair (χ, F) where χ is the vertex set of K v and F is a family of edge-disjoint subgraphs isomorphic to G in K v . In this paper, the maximum number of subgraphs in a ( v, G,1)-packing is determined for six graphs G of five vertices.

On nesting of G-decompositions of λKv where G has four nonisolated vertices or less

The complete multigraph λK v is said to have a G-decomposition if it is the union of edge disjoint subgraphs of K v each of them isomorphic to a fixed graph G. The spectrum problem for G-decompositions of λK v that have a nesting was first considered in the case G= K 3 by Colbourn and Colbourn (Ars Combin. 16 (1983) 27–34) and Stinson (Graphs and Combin. 1 (1985) 189–191). For λ=1 and G= C m (the cycle of length m) this problem was studied in many papers, see Lindner and Rodger (in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory: A Collection of Surveys, Wiley, New York, 1992, p. 325–369), Lindner et al. (Discrete Math. 77 (1989) 191–203), Lindner and Stinson (J. Combin. Math. Combin. Comput. 8 (1990) 147–157) for more details and references. For λ=1 and G= P k (the path of length k−1) the analogous problem was considered in Milici and Quattrocchi (J. Combin. Math. Combin. Comput. 32 (2000) 115–127). In this paper we solve the spectrum problem of nested G-decompositions of λK v for all the graphs G having four nonisolated vertices or less, leaving eight possible exceptions.

A Near Packing of Two Graphs

Let G 1 and G 2 be two graphs on n vertices with Δ ( G 1 )= d 1 and Δ ( G 2 )= d 2 . A packing of G 1 and G 2 is a set L ={ H 1 , H 2 } such that H 1 ≅ G 1 , H 2 ≅ G 2 , and H 1 and H 2 are edge disjoint subgraphs of K n . B. Bollobás and S. E. Eldridge (1978, J. Combin. Theory Ser. B 25 , 105–124) made the following conjecture. If ( d 1 +1)·( d 1 +1)⩽ n +1, then there is a packing of G 1 and G 2 . The degree condition stated in this conjecture could not be weakened, since there exist two such graphs for which ( d 1 +1)·( d 2 +1)= n +2, and no packing is possible. N. Sauer and J. Spencer (1978, J. Combin. Theory Ser. B 25 , 295–302) proved that if d 1 · d 2 < 1 2 n then there is a packing of G 1 and G 2 . In this paper, a generalization and extension of the Sauer–Spencer result is proven. We define a near packing of degree d of two graphs of order n as a generalization of a packing. In a near packing of degree d , the copies of the two graphs may overlap so that the maximum degree of the subgraph defined by the edges common to both copies is d . Thus a near packing of degree 0 is a packing. The result can then be stated as follows. Let a ∈ R + . If d 1 · d 2 ⩽ an , then there exists a near packing of degree d of G 1 and G 2 for some d <2 a . Furthermore, if ( d 1 +1)·( d 2 +1)⩽ n +1, then the maximum degree d of the subgraph defined by the edges common to both the copy of G 1 and the copy of G 2 is at most 1 and its size is no larger than n 2 +1− d 1 − d 2 .

Packing and Decomposition of Graphs with Trees

Let H be a tree on h⩾2 vertices. It is shown that if n is sufficiently large and G=(V, E) is an n-vertex graph with δ(G)⩾⌊n/2⌋, then there are ⌊|E|/(h−1)⌋ edge-disjoint subgraphs of G which are isomorphic to H. In particular, if h−1 divides |E| then there is an H-decomposition of G. This result is best possible as there are infinitely many examples of trees on h vertices and graphs G with m(h−1) edges, δ(G)⩾⌊n/2⌋−1, for which G has no H-decomposition.

Decompositions to Degree-Constrainded Subgraphs Are Simply Reducible to Edge-Colorings

The degree-constrained subgraphs decomposition problem, such as anf-coloring, anf-factorization, and a [g, f]-factorization, is to decompose a given graphG=(V, E) to edge-disjoint subgraphs degree-constrained by integer-valued functionsfandgonV. In this paper we show that the problem can be simply reduced to the edge-coloring problem in polynomial-time. That is, for any positive integerk, we give a polynomial-time transformation ofGto a new graph such thatGcan be decomposed to at mostkdegree-constrained subgraphs if and only if the new graph can be edge-colored withkcolors.

Construction of self-complementary graphs

Given a graph H with 2p vertices, the necessary and sufficient condition for the existence of a self-complementary graph G of order 4p containing a pair of edge disjoint subgraphs H1 and H2 such that H ≅ H1 and H ⋍ H2 is presented. An algorithm that generates all such self-complementary graphs is also presented.

On the Decomposition of Cayley Color Graphs into Isomorphic Oriented Trees

We prove that if Δ is a minimal generating set for a nontrivial group Γ and T is an oriented tree having | Δ| edges, then the Cayley color graph D Δ ( Γ) can be decomposed into Absolute value of | Γ| edge-disjoint subgraphs, each of which is isomorphic to T; we say that D Δ ( Γ) is T-decomposable. This result is extended to obtain a result concerning H-decompositions of Cayley graphs for weakly connected oriented graphs H. The first result is then used to derive several theorems concerning decompositions of Cayley color graphs into prescribed families of oriented trees. Applications of some of these theorems to the verification of statements about decompositions of the n-dimensional hypercube Q n are also discussed.

IEEE Trans. Reliab.,

A stochastic network is considered the underlying deterministic graph of which is assumed to be undirected and connected. This graph is split into two edge—disjoint subgraphs by a given separating vertex set. Reliability analysis is carried out for the corresponding stochastic subgraphs and the results are combined to obtain a formula of the K—terminal reliability of the stochastic network. Special cases are discussed in detail and numerical examples are presented.

Parity results on connected ƒ-factors

Let G be a connected graph with vertex set V and let d( ν) denote the degree of a vertex νϵV. For ƒ a mapping from V to the positive integers, an ƒ-factor is a spanning subgraph having degree ƒ(ν) at vertex ν. In this paper we extend the parity results of Thomason [2] on Hamiltonian circuits to connected ƒ-factors. (A Hamiltonian circuit is a connected 2-factor.) We show that if ƒ(ν) and d( ν) have opposite parity for all νϵV then for any given subgraph C there is an even number of connected ƒ-factors having C as a cotree. Let ƒ 1 and ƒ 2 be any mappings from V to the positive integers that partition d, i.e., d(ν) = ƒ 1(ν) +ƒ 2(ν) for all νϵV. Let C 1 and C 2 be any pair of edge disjoint subgraphs. We also show in this paper that the number of decompositions of G into a connected ƒ 1- factor having C 1 as a cotree and a connected ƒ 2- factor having C 2 as a cotree is even.

A decomposition of complete bipartite graphs into edge-disjoint subgraphs with star components

A subgraph F of a graph G is called a star-subgraph if each component of F is a star. In this note we show that the minimum number of star-subgraphs that partition the edges of K 2 n −1,2 n−1 or K 2 n,2 n , respectively, is n + 2.

Randomly decomposable graphs

A nonempty graph G is randomly H-decomposable if every family of edge-disjoint subgraphs of G, each subgraph isomorphic to H, can be extended to an H-decomposition of G. A characterization of those randomly H-decomposable graphs is given whenever H has two edges. Some related questions are discussed.

Resolvable path designs

A resolvable (balanced) path design, RBPD( v, k, λ) is the decomposition of λ copies of the complete graph on v vertices into edge-disjoint subgraphs such that each subgraph consists of v k vertex-disjoint paths of length k − 1 ( k vertices). It is shown that an RBPD( v, 3, λ) exists if and only if v ≡ 9 (modulo 12/gcd(4, λ)). Moreover, the RBPD( v, 3, λ) can have an automorphism of order v 3 . For k > 3, it is shown that if v is large enough, then an RBPD( v, k, 1) exists if and only if v ≡ k 2 (modulo lcm(2 k − 2, k)). Also, it is shown that the categorical product of a k-factorable graph and a regular graph is also k-factorable. These results are stronger than two conjectures of P. Hell and A. Rosa

On the Decomposition of Graphs

In this paper, we study the decompositions of a graph G into edge-disjoint subgraphs all of which belong to a specified class of graphs $\mathcal{H}$. Let $\alpha (G;\mathcal{H})$ denote the minimum value of the total sum of the sizes of subgraphs in $\mathcal{H}$ into which G can be decomposed, taken over all such decompositions of G. Let $\alpha (n;\mathcal{H})$ denote the maximum value of $\alpha (G;\mathcal{H})$ over all graphs G with n vertices. In this paper, we settle a conjecture of Katona and Tarján by showing \[ \alpha (n;\mathcal{K}) = \left\lfloor {n^2 /2} \right\rfloor ,\] where $\mathcal{K}$ denotes the set of all complete graphs. Moreover, we show that the complete bipartite graph G on $\lfloor n/2 \rfloor $ and $\lceil n/2 \rceil $ vertices is the only graph with $\alpha (G;\mathcal{K}) = \alpha (n;\mathcal{K})$.

G-decomposition of K n, where G has four vertices or less

The complete graph K n , is said to have a G-decomposition if it is the union of edge disjoint subgraphs each isomorphic to G. The set of values of n for which K n has a G-decomposition is determined if G has four vertices or less.

Decomposition of the n-Dimensional Lattice-Graph into Hamiltonian Lines

A graph G consists, for the purposes of this paper, of two disjoint sets V(G), E(G), whose elements are called vertices and edges respectively of G, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices (called its end-vertices) which the edge is said to join, and whereby no two vertices are joined by more than one edge. An edge γ and vertex ξ are incident if ξ is an end-vertex of γ. A monomorphism [isomorphism] of a graph G into [onto] a graph H is a one-to-one function φ from V(G)∪E(G) into [onto] V(H)∪E(H) such that φ(V(G))⊂V(H), φ(E(G))⊂E(H) and an edge and vertex of G are incident in G if and only if their images under φ are incident in H. G and H are isomorphic (in symbols, G ≅ H) if there exists an isomorphism of G onto H. A subgraph of G is a graph H such that V(H) ⊂ V(G), E(H)⊂E(G) and an edge and vertex of H are incident in H if and only if they are incident in G; if V(H) = V(G), H is a spanning subgraph. A collection of graphs are edge-disjoint if no two of them have an edge in common. A decomposition of G is a set of edge-disjoint subgraphs of G which between them include all the edges and vertices of G. Ln is a graph whose vertices are the lattice points of n-dimensional Euclidean space, two vertices A and B being joined by an edge if and only if AB is of unit length (and therefore necessarily parallel to one of the co-ordinate axes). An endless Hamiltonian line of a graph G is a spanning subgraph of G which is isomorphic to L1. The object of this paper is to prove that Ln is decomposable into n endless Hamiltonian lines, a result previously established (1) for the case where n is a power of 2.