Complete Multigraph Research Articles

Decomposition of Complete Multigraph into Wheel Graphs for Cyclic Triple System

Let 𝑛 be positive integer and 𝑊𝑛 be a wheel graph of order 𝑛. In this paper, we construct a new decomposition of 8𝐾𝑣 into wheel graphs for 𝑣 ≡ 2 (mod 12). Then, new cyclic triple system will be defined to arrange 𝑣 × 2(𝑣 − 1) triples satisfying certain criteria. In this development, the decomposition of 8𝐾𝑣 will be used to construct a cyclic 12-fold triple system of order 𝑣 ≡ 2 (mod 12).

MGRank: A keyword extraction system based on multigraph GoW model and novel edge weighting procedure

Keyword extraction is the process of extracting the most descriptive words from a textual document. State-of-the-art graph-based keyword extraction systems generally represent text documents using a simple graph called a graph-of-words (GoW), based on the sliding window concept. This representation of a text document requires determining a proper window size, models the document on a local scale, and allows the establishment of a single relation between two candidate keywords. In this study, we address these problems and propose a keyword extraction system called MGRank which uses a complete multigraph structure to build a GoW model to represent a text document. The completeness property of the proposed GoW model provides a means to represent a document globally and eliminates the need to determine the window-size parameter. Parallel edges allow the establishment of multiple relations between candidate keywords. In this study, we also propose a new edge-weighting procedure based on the positional distance of candidate keywords. To evaluate the performance of MGRank, we performed experiments on seven benchmark datasets and compared the results with those of six baseline methods. The experimental results show that MGRank outperforms the baseline methods statistically in precision, recall, and F1-score in almost all cases. In terms of mean average precision and mean reciprocal rank, MGRank performs statistically better than node ranking-based and statistical baseline methods and achieves on-par results with topic-based baseline methods. Furthermore, the experimental results showed that MGRank extracted the most relevant keywords.

Decompositions of Complete Multigraphs into Cyclic Designs

Let  and  be positive integer,  denote a complete multigraph. A decomposition of a graph  is a set of subgraphs of  whose edge sets partition the edge set of . The aim of this paper, is to decompose a complete multigraph  into cyclic -cycle system according to specified conditions. As the main consequence, construction of decomposition of  into cyclic Hamiltonian wheel system, where , is also given. The difference set method is used to construct the desired designs.

Decompositions of complete multigraphs into stars of varying sizes

In 1979 Tarsi showed that an edge decomposition of a complete multigraph into stars of size m exists whenever some obvious necessary conditions hold. In 1992 Lonc gave necessary and sufficient conditions for the existence of an edge decomposition of a (simple) complete graph into stars of sizes m1,…,mt. We show that the general problem of when a complete multigraph admits a decomposition into stars of sizes m1,…,mt is NP-complete, but that it becomes tractable if we place a strong enough upper bound on max⁡(m1,…,mt). We determine the upper bound at which this transition occurs. Along the way we also give a characterisation of when an arbitrary multigraph can be decomposed into stars of sizes m1,…,mt with specified centres, and a generalisation of Landau's theorem on tournaments.

Decomposition of complete uniform multi‐hypergraphs into Berge paths and cycles

AbstractIn 2015, Bryant, Horsley, Maenhaut, and Smith, generalizing a well‐known conjecture by Alspach, obtained the necessary and sufficient conditions for the decomposition of the complete multigraph into cycles of arbitrary lengths, where I is empty, when is even and I is a perfect matching, when is odd. Moreover, Bryant in 2010, verifying a conjecture by Tarsi, proved that the obvious necessary conditions for packing pairwise edge‐disjoint paths of arbitrary lengths in are also sufficient. In this article, first, we obtain the necessary and sufficient conditions for packing edge‐disjoint cycles of arbitrary lengths in . Then, applying this result, we investigate the analogous problem of the decomposition of the complete uniform multihypergraph into Berge cycles and paths of arbitrary given lengths. In particular, we show that for every integer , and , can be decomposed into Berge cycles and paths of arbitrary lengths, provided that the obvious necessary conditions hold, thereby generalizing a result by Kühn and Osthus on the decomposition of into Hamilton Berge cycles.

GENERALISATIONS OF THE DOYEN–WILSON THEOREM

In 1973, Doyen and Wilson famously solved the problem of when a 3-cycle system can be embedded in another 3-cycle system. There has been much interest in the literature in generalising this result for m-cycle systems when m > 3. Although there are several partial results, including complete solutions for some small values of m and strong partial results for even m, this still remains an open problem. The main results of this thesis concern generalisations of the Doyen-Wilson Theorem for odd m-cycle systems and cycle decompositions of the complete graph with a hole. The complete graph of order v with a hole of size u, Kv - Ku, is constructed from the complete graph of order v by removing the edges of a complete subgraph of order u (where v ≥ u). For each odd m ≥ 3 we completely solve the problem of when an m-cycle system of order u can be embedded in an m-cycle system of order v, barring a finite number of possible exceptions. The problem is completely resolved in cases where u is large compared to m, where m is a prime power, or where m ≤ 15. In other cases, the only possible exceptions occur when v - u is small compared to m. This result is proved as a consequence of a more general result which gives necessary and sufficient conditions for the existence of an m-cycle decomposition of Kv - Ku in the case where u ≥ m - 2 and v - u ≥ m + 1 both hold. We prove that Kv - Ku can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, v - u ≥ 10, each cycle has length at most min(u,v - u), and the longest cycle is at most three times as long as the second longest. This complements existing results for cycle decompositions of graphs such as the complete graph, complete bipartite graph and complete multigraph. We obtain these cycle decomposition results by applying a cycle switching technique to modify cycle packings of Kv - Ku. The tools developed by cycle switching enable us to merge collections of short cycles to obtain longer cycles. The methodology therefore relies on first finding decompositions of various graphs into short cycles, then applying the merging results to obtain the required decomposition. Similar techniques have previously been successfully applied to the complete graph and the complete bipartite graph. These methods also have potential to be further developed for the complete graph with a hole as well as other graphs. We also give a complete solution to the problem of when there exists a packing of the complete multigraph with cycles of arbitrary specified lengths. The proof of this result relies on applying cycle switching to modify cycle decompositions of the complete multigraph obtained from known results. The results in this thesis make substantial progress toward generalising the Doyen-Wilson Theorem for arbitrary odd cycle systems and toward constructing cycle decompositions of the complete graph with a hole. However there still remain unsolved cases. Moreover, the cycle switching and base decomposition methods used to obtain these results give rise to several interesting open problems.

Indecomposable 1‐factorizations of the complete multigraph for every

AbstractA 1‐factorization of the complete multigraph is said to be indecomposable if it cannot be represented as the union of 1‐factorizations of and , where . It is said to be simple if no 1‐factor is repeated. For every and for every , we construct an indecomposable 1‐factorization of , which is not simple. These 1‐factorizations provide simple and indecomposable 1‐factorizations of for every and . We also give a generalization of a result by Colbourn et al., which provides a simple and indecomposable 1‐factorization of , where , , p prime.

Decompositions of complete multigraphs into cycles of varying lengths

We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

Internally Fair Factorizations and Internally Fair Holey Factorizations with Prescribed Regularity

Let $G$ be a multipartite multigraph without loops. Then $G$ is said to be internally fair if its edges are shared as evenly as possible among all pairs of its partite sets. An internally fair factorization of $G$ is an edge-decomposition of $G$ into internally fair regular spanning subgraphs. A holey factor of $G$ is a regular subgraph spanning all vertices but one partite set. An internally fair holey factorization is an edge-decomposition of $G$ into internally fair holey factors. In this paper, we settle the existence of internally fair (respectively, internally fair holey) factorizations of the complete equipartite multigraph into factors (respectively, holey factors) with prescribed regularity.

On K p,q -factorization of complete bipartite multigraphs

Let λK m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K p,q -factorization of λK m,n is a set of edge-disjoint K p,q -factors of λK m,n which partition the set of edges of λK m,n . When p = 1 and q is a prime number, Wang, in his paper [On K 1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K 1,q -factorization of K m,n and gave a sufficient condition for such a factorization to exist. In papers [K 1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK m,n to have a K p,q -factorization. As a special case, it is shown that the necessary condition for the K p,q -factorization of λK m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.

{C<sub>k</sub>, P<sub>k</sub>, S<sub>k</sub>} -Decompositions of Balanced Complete Bipartite Multigraphs

Let be a family of subgraphs of a graph G. An L-decomposition of G is an edge-disjoint decomposition of G into positive integer copies of Hi, where . Let Ck, Pk and Sk denote a cycle, a path and a star with k edges, respectively. For an integer , we prove that a balanced complete bipartite multigraph has a -decomposition if and only if k is even, and .

A note on m-factorizations of complete multigraphs arising from designs

Some new infinite families of simple, indecomposable $m$-factorizations of the complete multigraph $\lambda K_v$ are presented. Most of the constructions come from finite geometries.

A New Proof of Cayleyʼs Formula for Labeled Spanning Trees

Cayleysʼs formula for enumerating spanning trees in complete graphs is one of the main theorems in this topic. In this note based on a new recursive method for the enumeration of spanning trees and meanwhile enumerating spanning trees in a kind of complete multigraph, a new proof for the Cayleyʼs formula is given. A generalization of this formula is also presented.

On Heterogeneous Decompositions of Uniform Complete Multigraphs into Spanning Trees

Let be the order n uniform complete multigraph with edge multiplicity r. A spanning tree decomposition of partitions its edge set into a family J of edge-induced spanning trees. In a purely heterogeneous decomposition J no trees are isomorphic. Every order n tree occurs in a fully heterogeneous decomposition J. All trees have equal multiplicity in a balanced decomposition J. We show: (1) has 16 decomposition classes, four of which contain the 24 fully heterogeneous decompositions; (2) has 34 fully heterogeneous decomposition classes; exactly one lacks decompositions reducible to two decompositions; (3) when k ≥ 1, many balanced fully heterogeneous decompositions of reduce to “smaller” decompositions, but one such decomposition of is irreducible.

REAL TIME MULTIGRAPHS FOR COMMUNICATION NETWORKS: AN INTUITIONISTIC FUZZY MATHEMATICAL MODEL

Many problems of computer science, communication network, transportation systems, can be modeled into multigraphs (or graphs) and then can be solved. Nowadays, the networks are expanding very fast in huge volumes in terms of their nodes and the connecting links. For a given alive network, in many situations, its complete topology may not be always available to the communication systems at a given point of time because of the reason that few or many of its links (edges/arcs) may be temporarily disable owing to damage or attack or blockage upon them and of course they are under repair at that point of time. Such cases are now so frequent that it calls for rigorous attention of the researchers, in particular to those who are concerned with Quality of Service (QoS) while in a network. Even in most of the cases the cost parameters corresponding to its links are not crisp numbers, rather intuitionistic fuzzy numbers (or fuzzy numbers). Thus at any real time instant, the complete multigraph is not available but a submutigraph of it is available to the system for executing its communication or transportation activities. Under such circumstances, none of the existing algorithms on Shortest Path Problems (SPP) can work. In this study the authors propose a mathematical model for such types of multigraphs to be called by ‘Real Time Multigraphs’ (RT-multigraphs) in which all real time information (being updated every q quantum of time) are incorporated so that the communication/transportation system can be made very efficiently with optimal results. It is a kind of intuitionistic fuzzy mathematical model being the most generalized form of the crisp multigraphs. As a special case, RT-multigraphs reduce to the case of ‘RT-graphs’. Finally an intuitionistic fuzzy method is developed to solve the shortest path problem in a RT-Multigraph. As a special case the problem reduces to fuzzy shortest path problem in a RT-Multigraph.

Intuitionistic Fuzzy Real Time Multigraphs for Communication Networks: A Theoretical Model

Many problems of computer science, communication network, transportation systems, etc. can not be modeled into graphs, but into multigraphs only, and then can be easily solved. Nowadays, the networks are expanding very fast in huge volumes in terms of their nodes and links/arcs. For a given alive network, in many situations, its complete topology may not be always available to the communication systems at a given point of time because of the reason that few or many of its links/arcs may be temporarily disable owing to damage or external attack or blockage upon them, and of course they are under repair at that point of time. Besides that, in most of the cases the cost parameters corresponding to its links are not crisp numbers, rather intuitionistic fuzzy numbers (or fuzzy numbers). Thus at any real time instant, the complete multigraph is not available but a submutigraph of it is available to the system for executing its communication or packets transfer. There is no mathematical model available in the existing literature to represent such type of real time network. In this paper the authors propose a mathematical model for such types of multigraphs be called by ‘Real Time Multigraphs’ (RT-multigraphs) in which all real time information (being updated every q quantum of time) are incorporated so that the communication/transportation system can be made very efficiently with optimal results. It is a theoretical work, a kind of intuitionistic fuzzy mathematical model being the most generalized form of the crisp multigraphs.

Symmetric Hamilton cycle decompositions of complete multigraphs

Let n ≥ 3 and λ ≥ 1 be integers. Let λKn denote the complete multigraph with edge-multiplicity λ. In this paper, we show that there exists a symmetric Hamilton cycle decomposition of λK2m for all even λ ≥ 2 and m ≥ 2. Also we show that there exists a symmetric Hamilton cycle decomposition of λK2m − F for all odd λ ≥ 3 and m ≥ 2. In fact, our results together with the earlier results (by Walecki and Brualdi and Schroeder) completely settle the existence of symmetric Hamilton cycle decomposition of λKn (respectively, λKn − F , where F is a 1-factor of λKn) which exist if and only if λ(n − 1) is even (respectively, λ(n − 1) is odd), except the non-existence cases n ≡ 0 or 6 (mod 8) when λ = 1.

Cycle decompositions of complete multigraphs

It is shown that the obvious necessary conditions for the existence of a decomposition of the complete multigraph with n vertices and with λ edges joining each pair of distinct vertices into m-cycles, or into m-cycles and a perfect matching, are also sufficient. This result follows as an easy consequence of more general results which are obtained on decompositions of complete multigraphs into cycles of varying lengths. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:42-69, 2010

Edge-disjoint Open Trails in Complete Bipartite Multigraphs

Let r K a,b be a complete bipartite multigraph. We show a necessary and sufficient condition for a multigraph r K a,b to be arbitrarily decomposable into open trails. We extend the results obtained by Balister for complete graphs (Balister in Probab Comput 10:463–499, 2001). Moreover we show that a multigraph r K a,b with even r and a ≥ 3 or b ≥ 3 is arbitrarily decomposable into open and closed trails.

Packing paths in complete graphs

Let λ K n denote the complete graph of order n and multiplicity λ. We prove Tarsi's conjecture [M. Tarsi, Decomposition of a complete multigraph into simple paths: Nonbalanced handcuffed designs, J. Combin. Theory Ser. A 34 (1983) 60–70] that for any positive integers n, λ and t, and any sequence m 1 , m 2 , … , m t of positive integers, it is possible to pack t pairwise edge-disjoint paths of lengths m 1 , m 2 , … , m t in λ K n if and only if m i ⩽ n − 1 for i = 1 , 2 , … , t and m 1 + m 2 + ⋯ + m t ⩽ λ n ( n − 1 ) 2 .