On the ascending subgraph decomposition problem for bipartite graphs

In 1987, Alavi, Boals, Chartrand, Erdos, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). Though different classes of graphs have been shown to have an ASD, the conjecture remains open. In this paper we investigate the similar problem for digraphs. In particular, we will show that any orientation of a compete balanced tripartite graph has an ASD.

Let t and q be positive integers that satisfy C ( t + 1 , 2) ≤ q < C ( t + 2 , 2) and let G be a simple and finite graph of size q . G is said to have ascending subgraph decomposition (ASD) if G can be decomposed into t subgraphs H 1 ,H 2 , … ,H t without isolated vertices such that H i is isomorphic to a proper subgraph of H i +1 for 1 ≤ i ≤ t - 1 , where { E ( H 1 ) , … ,E ( H t ) } is a partition of E ( G ) . A graph that admits an ascending subgraph decomposition is called an ASD graph. In this paper, we introduce a new type of magic labeling based on the notion of ASD. Let G be an ASD graph and f : V ( G ) ∪ E ( G ) →{ 1 , 2 , … , | V ( G ) | + | E ( G ) |} be a bijection. The weight of a subgraph H i (1 ≤ i ≤ n ) is w ( H i ) = ∑ v ∈ V ( H i ) f ( v ) + ∑ e ∈ E ( H i ) f ( e ) . If the weight of each ascending subgraph is constant, say w ( H i ) = k , ∀ 1 ≤ i ≤ t , then f is called an ASD-magic labeling of G and G is called an ASD-magic graph. We present general properties of ASD-magic graphs and characterize certain classes of them.

Star forests, dominating sets and Ramsey-type problems

For any positive integer k, the total k-domatic partition problem is to partition the vertices of a graph G into k pairwise disjoint total dominating sets. In this paper, we study the problem for planar graphs, chordal bipartite graphs, convex bipartite graphs, and bipartite permutation graphs. We show that the total 3-domatic partition problem on planar graphs is NP-complete. Moreover, we give an alternative algorithm to solve the total k-domatic partition problem for chordal bipartite graphs with weak elimination orderings, and adapt it to solve the problem in linear time for bipartite permutation graphs and convex bipartite graphs even if Gamma-free forms of the adjacency matrices of the considered graphs are not given.

We consider two generalizations of the edge coloring problem in bipartite graphs. The first problem we consider is the weighted bipartite edge coloring problem where we are given an edge-weighted bipartite graph G = (V,E) with weights w:E→[0,1]. The task is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring of the weighted graph is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. We give a polynomial time algorithm for the weighted bipartite edge coloring problem which returns a proper weighted coloring using at most ⌈2.25 n⌉ colors where n is the maximum total weight incident at any vertex. This improves on the previous best bound of Correa and Goemans [5] which returned a coloring using 2.557n + o(n) colors. The second problem we consider is the Balanced Decomposition of Bipartite graphs problem where we are given a bipartite graph G = (V,E) and α 1,...,α k ∈ (0,1) summing to one. The task is to find a partition E 1,..., E k of E such that \(deg_{E_i}(v)\) is close to α i deg E (v) for each 1 ≤ i ≤ k and v ∈ V. We give an alternate proof of the result of Correa and Goemans [5] that there is a decomposition such that \(\lfloor\alpha_i deg_E(v) \rfloor -2 \leq deg_{E_i}(v) \leq \lceil \alpha_i deg_E(v) \rceil +2\) for each v ∈ V and 1 ≤ i ≤ k. Moreover, we show that the additive error can be improved from two to one if only upper bounds or only lower bounds on the degree are present. All our results hold also for bipartite multigraphs, and the decomposition results hold also for general graphs.KeywordsBipartite GraphWeight GraphEdge IncidentGeneral GraphEdge ColoringThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the NP-completeness of the perfect matching free subgraph problem

In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on $n$ vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in $n$ in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method.

We consider two problems: randomly generating labeled bipartite graphs with a given degree sequence and randomly generating labeled tournaments with a given score sequence. We analyze simple Markov chains for both problems. For the first problem, we cannot prove that our chain is rapidly mixing in general, but in the near-regular case, i.e., when all the degrees are almost equal, we give a proof of rapid mixing. Our methods also apply to the corresponding problem for general (nonbipartite) regular graphs, which was studied earlier by several researchers. One significant difference in our approach is that our chain has one state for every graph (or bipartite graph) with the given degree sequence; in particular, there are no auxiliary states as in the chain used by Jerrum and Sinclair. For the problem of generating tournaments, we are able to prove that our Markov chain on tournaments is rapidly mixing, if the score sequence is near-regular. The proof techniques we use for the two problems are similar. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14: 293–308, 1999

Tip decomposition has a pivotal role in mining cohesive subgraphs in bipartite graphs by computing the tip number of each vertex in accordance with the non-trivial motif butterfly ((2,2)-biclique). It has been a popular research topic with applications in document clustering, spam group detection, and analysis of affiliation networks. In such applications, the graphs are not only large, but they evolve quickly with new edges being continuously added/deleted. While existing centralized techniques could solve the tip decomposition problem for static bipartite graphs, they are not efficient for maintaining the tip numbers of vertices on large-scale graphs. In this paper, we study butterfly analysis problems on bipartite graphs in a distributed environment with the vertex-centric model. We first extend a centralized butterfly counting algorithm to a distributed version, called DBCA. An ingenious message aggregation strategy is designed to reduce massive redundant messaging and avoid the memory overflow problem while processing large-scale graphs. Based on the results of DBCA, we develop a distributed tip decomposition algorithm (DTDA) to get the tip number of each vertex in parallel. Finally, to maintain the tip numbers of vertices efficiently while graphs evolve over time, we explore a distributed tip maintenance algorithm (DTMA) along with a novel task-split strategy. Specifically, for an updated edge (insertion/deletion), several sub-tasks will be generated in line with the topology structure of the original bipartite graph. To our best knowledge, this is the first study to process the butterfly analysis problems in a distributed environment. In addition, comprehensive experiments have been conducted on real-world bipartite graphs. The experiment results demonstrate that our proposed algorithms are efficient and scalable.

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\max}$ satisfies $r^4 k_{\max}^4(k_{\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.

On the complexity of the flow coloring problem

A graph theoretical representation for a class of interconnection networks is suggested. The idea is based on a definition of orthogonal binary vectors and leads to a construction rule for a class of orthogonal graphs. An orthogonal graph is first defined as a set of 2/sup m/ nodes, which in turn are linked by 2/sup m-n/ edges for every link model defined in an integer set Q*. The degree and diameter of an orthogonal graph are determined in terms of the parameters n, m, and the number of link modes defined in Q*. Routing in orthogonal graphs is shown to reduce to the node covering problem in bipartite graphs. The proposed theory is applied to describe a number of well-known interconnection networks such as the binary m-cube and spanning-bus meshes. Multidimensional access (MDA) memories are also shown as examples of orthogonal shared memory multiprocessing systems. Finally, orthogonal graphs are applied to the construction of multistage interconnection networks. Connectivity and placement rules are given and shown to yield a number of well-known networks. >

A polynomial algorithm for the extendability problem in bipartite graphs

Online request server matching

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (J Comb Theory Ser B 73:130–145, 1998), and the tractability of the problem is ascribed to total dual integrality in a linear programming formulation by Schrijver (Ann Discret Math 16:261–280, 1982). Another view on the tractability of this problem is afforded by a valuated matroid intersection formulation by Takazawa (Jpn J Ind Appl Math 29:561–573, 2012). In the present paper, we discuss the relationship between these two formulations for the shortest bibranching problem. We first demonstrate that the valuated matroid intersection formulation can be derived from the linear programming formulation through the Benders decomposition, where integrality is preserved in the decomposition process and the resulting convex programming is endowed with discrete convexity. We then show how a pair of primal and dual optimal solutions of one formulation is constructed from that of the other formulation, thereby providing a connection between polyhedral combinatorics and discrete convex analysis.