Spanning Subgraph Research Articles

Minimum degree conditions for containing an [formula omitted]-regular [formula omitted]-connected spanning subgraph

We study optimal minimum degree conditions when an n-vertex graph G contains an r-regular r-connected spanning subgraph. We prove for r fixed and n large the condition to be δ(G)≥n+r−22 when nr≡0(mod2). This answers a question of M. Kriesell.

Disconnected spanning subgraphs of paths in the join products with cycles

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main aim of this paper is to give the crossing numbers of the join products of the cycles Cn on n vertices with all nonempty disconnected spanning subgraphs of the paths P5 and P6 on five and six vertices, respectively. Recently, this problem was solved for the nonempty disconnected spanning subgraphs of the paths on three and four vertices by Klešč.

The complexity of finding low chromatic spanning sub(di)graphs with prescribed connectivity properties

As usual λ(G) denotes the edge-connectivity of the graph G. It was shown in [2] that every graph G contains a spanning (λ(G)+1)-partite subgraph H such that λ(H)=λ(G) and one can find such a spanning subgraph in polynomial time. We determine the complexity of deciding, for given positive integers r,k whether a graph contains a spanning r-colourable subgraph which is k-edge-connected. We show that the problem is polynomially solvable when r>k and NP-complete otherwise. In fact, combined with the result from [2] above, this means that the problem is polynomially solvable precisely when r is such that every k-edge-connected graph has a spanning r-colourable subgraph which is k-edge-connected.One can show that all graphs whose edge set decomposes into k edge-disjoint spanning trees are 2k-colourable. We consider the problem of deciding whether a given graph G has a collection of k edge-disjoint spanning trees whose union forms an r-colourable spanning subgraph H of G. We show that this problem is polynomially solvable when r≥2k and NP-complete for all other values of r. We also determine the complexity of the analogous problem of deciding whether a digraph D has a collection of k arc-disjoint out-branchings such that the spanning subdigraph formed by the union of the arcs in the branchings is r-colourable.

A new approximation algorithm for the minimum 2-edge-connected spanning subgraph problem

We present a new approximation algorithm for the minimum 2-edge-connected spanning subgraph problem. Its approximation ratio is 43, which matches the current best ratio. The approximation ratio of the algorithm is 65 on subcubic graphs, which is an improvement upon the previous best ratio of 54. The algorithm is a novel extension of the primal-dual schema, which consists of two distinct phases. Both the algorithm and the analysis are much simpler than those of the previous approaches.

Path-factor critical covered graphs and path-factor uniform graphs

A path-factor in a graphGis a spanning subgraphFofGsuch that every component ofFis a path. Letdandnbe two nonnegative integers withd ≥ 2. AP≥d-factor ofGis its spanning subgraph each of whose components is a path with at leastdvertices. A graphGis called aP≥d-factor covered graph if for anye ∈ E(G),Gadmits aP≥d-factor containinge. A graphGis called a (P≥d, n)-factor critical covered graph if for anyN ⊆ V(G) with |N| =n, the graphG − Nis aP≥d-factor covered graph. A graphGis called aP≥d-factor uniform graph if for anye ∈ E(G), the graphG − eis aP≥d-factor covered graph. In this paper, we verify the following two results: (i) An (n + 1)-connected graphGof order at leastn + 3 is a (P≥3, n)-factor critical covered graph ifGsatisfiesδ(G) > (α(G)+2n+3)/2; (ii) Every regular graphGwith degreer ≥ 2 is aP≥3-factor uniform graph.

SelectiveEC: Towards Balanced Recovery Load on Erasure-Coded Storage Systems

Erasure coding (EC) has been commonly used to offer high data reliability with low storage cost. Upon failures, the lost blocks are recovered in batches. Due to the limited number of stripes, the data layout within a batch is non-uniform. Together with the random selection of source and replacement nodes for recovery tasks, the recovery workload among live nodes is skewed within a batch, which severely slows down failure recovery. To solve this problem, We present SelectiveEC, a new recovery task scheduling module that provides provable network traffic and recovery load balancing for large-scale EC-based storage systems. It relies on bipartite graphs to model the recovery traffic among live nodes. Then, it intelligently selects tasks to form batches and carefully determines where to read source blocks or to store recovered ones, using theories such as a perfect or maximum matching and <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -regular spanning subgraph. SelectiveEC supports single-node failure and multi-node failure recovery, and can be deployed in both homogeneous and heterogeneous network environments. We implement SelectiveEC in HDFS, and evaluate its recovery performance in a local cluster of 18 nodes and AWS EC2 of 50 virtual machine instances. SelectiveEC increases the recovery throughput by up to <inline-formula><tex-math notation="LaTeX">$30.68\%$</tex-math></inline-formula> compared with state-of-the-art baselines in homogeneous network environments. It further achieves <inline-formula><tex-math notation="LaTeX">$1.32\times$</tex-math></inline-formula> recovery throughput and <inline-formula><tex-math notation="LaTeX">$1.23\times$</tex-math></inline-formula> benchmark throughput of HDFS on average in heterogeneous network environments, due to the straggler avoidance by the balanced scheduling.

Packing Trees into 1-planar Graphs

We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, \dots, G_k$, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each $G_i$ is a tree and $k=3$. We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with $n \geq 12$ vertices admits a 1-planar packing, while such a packing does not exist if $n \leq 10$.

Sparse graphs and an augmentation problem

For two integers k>0 and ell , a graph G=(V,E) is called (k,ell )-tight if |E|=k|V|-ell and i_G(X)le k|X|-ell for each Xsubseteq V for which i_G(X)ge 1, where i_G(X) denotes the number of induced edges by X. G is called (k,ell )-redundant if G-e has a spanning (k,ell )-tight subgraph for all ein E. We consider the following augmentation problem. Given a graph G=(V,E) that has a (k,ell )-tight spanning subgraph, find a graph H=(V,F) with the minimum number of edges, such that Gcup H is (k,ell )-redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is (k,ell )-tight. For general inputs, we give a polynomial algorithm when kge ell and show the NP-hardness of the problem when k<ell . Since (k,ell )-tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.

Generalized Hadamard Matrices and 2-Factorization of Complete Graphs

Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor of a graph is a spanning subgraph of which is not totally disconnected. An - factor is an - regular spanning subgraph of and is -factorable if there are edge-disjoint -factors such that . We shall refer as an -factorization of a graph . In this research we consider -factorization of complete graph. A graph with vertices is called a complete graph if every pair of distinct vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group of prime order , a square matrix of order all of whose elements are the root of unity is called a generalized Hadamard matrix if , where is the conjugate transpose of matrix and is the identity matrix of order . In the present work, generalized Hadamard matrices over a cyclic group have been considered. We prove that the factorization is possible for in the case of the limitation 1, namely, If an edge belongs to the factor , then the and entries of the corresponding generalized Hadamard matrix should be different in the row. In Particular, number of rows in the generalized Hadamard matrices is used to form -factorization of complete graphs. We discuss some illustrative examples that might be used for studying the factorization of complete graphs.

On the Construction of Partial Balanced Incomplete Block Designs Using Spanning Subgraphs

On the Construction of Partial Balanced Incomplete Block Designs Using Spanning Subgraphs

Circular Intensely Orthogonal Double Cover Design of Balanced Complete Multipartite Graphs

In this paper, we generalize the orthogonal double covers (ODC) of Kn,n as follows. The circular intensely orthogonal double cover design (CIODCD) of X=Kn,n,…,n︸m is defined as a collection T={G00,G10,…,G(n−1)0}∪{G01,G11,…,G(n−1)1} of isomorphic spanning subgraphs of X such that every edge of X appears twice in the collection T,E(Gi0)∩E(Gj0)=E(Gi1)∩E(Gj1)=0,i≠jand E(Gi0)∩E(Gj1)=λ=m2,i,j∈ℤn. We define the half starters and the symmetric starters matrices as constructing methods for the CIODCD of X. Then, we introduce some results as a direct application to the construction of CIODCD of X by the symmetric starters matrices.

Balanced Equi-$n$-Squares

We define a $d$-balanced equi-$n$-square $L=(l_{ij})$, for some divisor $d$ of $n$, as an $n \times n$ matrix containing symbols from $\mathbb{Z}_n$ in which any symbol that occurs in a row or column, occurs exactly $d$ times in that row or column. We show how to construct a $d$-balanced equi-$n$-square from a partition of a Latin square of order $n$ into $d \times (n/d)$ subrectangles. In graph theory, $L$ is equivalent to a decomposition of $K_{n,n}$ into $d$-regular spanning subgraphs of $K_{n/d,n/d}$. We also study when $L$ is diagonally cyclic, defined as when $l_{(i+1)(j+1)}=l_{ij}+1$ for all $i,j \in \mathbb{Z}_n$, which correspond to cyclic such decompositions of $K_{n,n}$ (and thus $\alpha$-labellings).&#x0D; We identify necessary conditions for the existence of (a) $d$-balanced equi-$n$-squares, (b) diagonally cyclic $d$-balanced equi-$n$-squares, and (c) Latin squares of order $n$ which partition into $d \times (n/d)$ subrectangles. We prove the necessary conditions are sufficient for arbitrary fixed $d \geq 1$ when $n$ is sufficiently large, and we resolve the existence problem completely when $d \in \{1,2,3\}$.&#x0D; Along the way, we identify a bijection between $\alpha$-labellings of $d$-regular bipartite graphs and what we call $d$-starters: matrices with exactly one filled cell in each top-left-to-bottom-right unbroken diagonal, and either $d$ or $0$ filled cells in each row and column. We use $d$-starters to construct diagonally cyclic $d$-balanced equi-$n$-squares, but this also gives new constructions of $\alpha$-labellings.

On balanced bipartitions of graphs

Bollobás and Scott conjectured that every graph G has a balanced bipartite spanning subgraph H such that for each for each In this paper, we consider the contrary side and show that every graphic sequence has a realization G which admits a balanced bipartite spanning subgraph H such that for each and we show that the bound is sharp.

A rainbow blow-up lemma for almost optimally bounded edge-colourings

Abstract A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.

Infinitesimal Rigidity in Normed Planes

We prove that a graph has an infinitesimally rigid placement in a non-Euclidean normed plane if and only if it contains a $(2,2)$-tight spanning subgraph. The method uses an inductive construction based on generalised Henneberg moves and the geometric properties of the normed plane. As a key step, rigid placements are constructed for the complete graph $K_4$ by considering smoothness and strict convexity properties of the unit ball.

Monochromatic [formula omitted]-edge-connection colorings of graphs

A path in an edge-colored graph G is called monochromatic if any two edges on the path have the same color. For k≥2, an edge-colored graph G is said to be monochromatic k-edge-connected if every two distinct vertices of G are connected by at least k edge-disjoint monochromatic paths, and G is said to be uniformly monochromatic k-edge-connected if every two distinct vertices are connected by at least k edge-disjoint monochromatic paths such that all edges of these k paths are colored with a same color. We use mck(G) and umck(G) to denote the maximum number of colors that ensures G to be monochromatic k-edge-connected and, respectively, G to be uniformly monochromatic k-edge-connected. In this paper, we first conjecture that for any k-edge-connected graph G, mck(G)=e(G)−e(H)+⌊k2⌋, where H is a minimum k-edge-connected spanning subgraph of G. We verify the conjecture for k=2. We also prove the conjecture for G=Kk+1 and G=Kk,n with n≥k≥3. When G is a minimal k-edge-connected graph, we give an upper bound of mck(G), i.e., mck(G)≤k−1. For the uniformly monochromatic k-edge-connectivity, we prove that for all k, umck(G)=e(G)−e(H)+1, where H is a minimum k-edge-connected spanning subgraph of G.

Enumerating Highly-Edge-Connected Spanning Subgraphs

Enumerating Highly-Edge-Connected Spanning Subgraphs

Anti-Ramsey Problems in Complete Bipartite Graphs for t Edge-Disjoint Rainbow Spanning Subgraphs: Cycles and Matchings

A subgraph H of an edge-colored graph G is rainbow if all of its edges have different colors. The anti-Ramsey number is the maximum number of colors in an edge-coloring of G with no rainbow copy of H. Originally a complete graph was considered as G. In this paper, we consider a complete bipartite graph as the host graph and discuss some results for the graph H being hamiltonian cycle and perfect matching. Let $$c(K_{p,p},t)$$ and $$m(K_{p,p},t)$$ be the maximum number of colors in an edge-coloring of the complete bipartite graph $$K_{p,p}$$ not having t edge-disjoint rainbow hamiltonian cycles and perfect matchings, respectively. We prove that $$c(K_{p,p},t)=p^2-p+t$$ for $$t \ge 2$$ , $$p\ge 4t-1$$ and $$m(K_{p,p},t)=p^2-2p+t+1$$ for $$t \ge $$ 2, $$p\ge 2t+8$$ .

On problems about judicious bipartitions of graphs

Bollobás and Scott [5] conjectured that every graph G has a balanced bipartite spanning subgraph H such that for each v∈V(G), dH(v)≥(dG(v)−1)/2. In this paper, we show that every graphic sequence has a realization for which this Bollobás-Scott conjecture holds, confirming a conjecture of Hartke and Seacrest [8]. On the other hand, we give an infinite family of counterexamples to this Bollobás-Scott conjecture, which indicates that ⌊(dG(v)−1)/2⌋ (rather than (dG(v)−1)/2) is probably the correct lower bound. We also study bipartitions V1,V2 of graphs with a fixed number of edges. We provide a (best possible) upper bound on e(V1)λ+e(V2)λ for any real λ≥1 (the case λ=2 is a question of Scott [12]) and answer a question of Scott (Problem 3.9 in [12]) on max⁡{e(V1),e(V2)}.

Rainbow matchings in Dirac bipartite graphs

We show the existence of rainbow perfect matchings in μn‐bounded edge colorings of Dirac bipartite graphs, for a sufficiently small μ &gt; 0. As an application of our results, we obtain several results on the existence of rainbow k‐factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree.