Multipartite Graphs Research Articles

The maximum number of complete multipartite subgraphs in graphs with given circumference or matching number

The circumference c(G) of a graph G is the length of a longest cycle in G and the matching number α′(G) is the maximum size of a matching in G. In 1959, Erdős and Gallai determined the maximum size of graphs with given c(G) or α′(G). Let Kr1,…,rs denote the complete multipartite graph with class sizes r1,…,rs and δ(G) be the minimum degree of G. In this paper, we determine the maximum number of copies of Kr1,…,rs in a 2-connected n-vertex graph G with given c(G) and δ(G)≥k. The corresponding problem for an n-vertex graph G with given α′(G) and δ(G)≥k is also addressed. As a corollary of our main results, we determine the maximum number of copies of Kr1,…,rs in an n-vertex graph with given c(G) or α′(G). In addition, the corresponding problems for graphs with given detour order are solved as well.

A conditional random field recommendation method based on tripartite graph

Recommender System (RS) has generated widespread attention with the aim of expanding different items. Among graph-based recommendation methods, the tripartite graph can better manage data sparsity and cold start, while improving the metrics of various recommendations such as recall, precision, and diversity. Existing tripartite graph-based methods encounter numerous challenges, including mitigating data sparsity, improving diversity, and capturing potential user preferences via social relations. To address these challenges, a Conditional Random Field based on Tripartite Graph (CRF-TG) is proposed. The tripartite graph consists of the user, item, and trust level. The method can mine potentially similar users, create probabilistic models based on TG, and uncover potential user preferences. Moreover, to mine the users with similar preferences outside the social relationship, the random walk method is used to test CRF-TG. Experiments are designed to verify the validity of CRF-TG. Compared to the others considered methods, CRF-TG gives a 15% increase on average in performance indicators such as diversity, recall, and F1.

Two-geodesic transitive graphs of order pn with n ≤ 3

A vertex triple (u,v,w) of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive if its automorphism group is transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order pn is given for each prime p and n≤3. It turns out that all such graphs consist of three small graphs: the complete bipartite graph K4,4 of order 8, the Schläfli graph of order 27 and its complement, and fourteen infinite families: the cycles Cp,Cp2 and Cp3, the complete graphs Kp,Kp2 and Kp3, the complete multipartite graphs Kp[p], Kp[p2] and Kp2[p], the Hamming graph H(2,p) and its complement, the Hamming graph H(3,p), and two infinite families of normal Cayley graphs on the extraspecial group of order p3 and exponent p.

BILANGAN KETERHUBUNGAN TITIK PELANGI BEBERAPA KELAS GRAF

A graph G is called a rainbow vertex connected if every two vertices G are connected by a rainbow path, that is, a path whose all the internal vertices are of a different color. The rainbow vertex connection number of graph G denoted by rvc(G) is the minimum number of colors used to color all vertices by G such that the graph G is connected to rainbow vertex. The rainbow vertex connection number in a graph will not be less than the diameter of the graph minus one. The rainbow vertex connection number discussed in this article for various classes of graphs include complete graph Kn, complete bipartite graph Km,n , wheel graph Wn , two-layer wheel graph Wn2, complete multipartite graph Kn1,n2,...,nt , path Pn, comb graph GSn, graph , graph , graph , graph .
 Keywords: graph, vertex coloring, rainbow vertex connection number.

Resource Allocation Strategy Based On Tripartite Graph in Vehicular Social Networks

Owing to the limited spectrum of vehicular social networks and the fragmented distribution of caching resources, problems such as low vehicle data transmission rate, poor service quality, and low service content delivery efficiency of streaming media are encountered. To solve these problems, we propose a resource allocation strategy based on tripartite graph in vehicular social networks (RATG). This strategy establishes a mobile vehicular social network model based on vehicle mobility and social similarity. To maximize the content delivery efficiency and transmission rate, one-to-one stable matching based on mobile social connections and one-to-one stable matching based on channel state are established. The proposed strategy incorporates a heuristic optimization algorithm based on the maximum spanning tree, such that the matching results of the two tripartite graphs can reach the approximate global optimal simultaneously and further improve the service quality of vehicles. The simulation results show that the content delivery efficiency of the RATG strategy is 6.05%, 5.00%, and 10.64% higher than those of the randomized resources allocation strategy, local search strategy, and local ratio strategy, respectively, and the average transmission rate is at least 2.7% higher for the RATG strategy, relative to those of the other strategies.

Complete Multipartite Graphs Decompositions Using Mutually Orthogonal Graph Squares

Complete Multipartite Graphs Decompositions Using Mutually Orthogonal Graph Squares

Complete forcing numbers of complete and almost-complete multipartite graphs

Complete forcing numbers of complete and almost-complete multipartite graphs

A generalization of group divisible t $t$‐designs

AbstractCameron defined the concept of generalized ‐designs, which generalized ‐designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized ‐designs and group divisible ‐designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible ‐designs, and then point out close connections with various well‐known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible ‐designs and almost completely determine their existence for and small block sizes.

Mutual witness Gabriel drawings of complete bipartite graphs

Let Γ be a straight-line drawing of a graph and let u and v be two vertices of Γ. The Gabriel disk of u,v is the disk having u and v as antipodal points. A pair 〈Γ0,Γ1〉 of vertex-disjoint straight-line drawings forms a mutual witness Gabriel drawing when, for i=0,1, any two vertices u and v of Γi are adjacent if and only if their Gabriel disk does not contain any vertex of Γ1−i. We characterize the pairs 〈G0,G1〉 of complete bipartite graphs that admit a mutual witness Gabriel drawing. The characterization leads to a linear time testing algorithm. We also show that when the pair 〈G0,G1〉 consists of two complete multi-partite graphs whose partition sets all have size greater than one, then the pair does not admit a mutual witness Gabriel drawing unless the pair is 〈K2,2,K2,2〉.

Extreme and statistical properties of eigenvalue indices of simple connected graphs

We analyze graphs attaining the extreme values of various spectral indices in the class of all simple connected graphs, as well as in the class of graphs which are not complete multipartite graphs. We also present results on density of spectral gap indices and its nonpersistency with respect to small perturbations of the underlying graph. We show that a small change in the set set of edges may result in a significant change of the spectral index like, e.g., the spectral gap or spectral index. We also present a statistical and numerical analysis of spectral indices of graphs of the order m≤10. We analyze the extreme values for spectral indices for graphs and their small perturbations. Finally, we present the statistical and extreme properties of graphs on m≤10 vertices.

On sufficient conditions for spanning structures in dense graphs

AbstractWe study structural conditions in dense graphs that guarantee the existence of vertex‐spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of Kühn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Schülke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore‐type degree conditions, Pósa‐type degree conditions, deficiency‐type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.

Reducing Graph Parameters by Contractions and Deletions

We consider the following problem: for a given graph G and two integers k and d, can we apply a fixed graph operation at most k times in order to reduce a given graph parameter π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} by at least d? We show that this problem is NP-hard when the parameter is the independence number and the graph operation is vertex deletion or edge contraction, even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document} and when restricted to chordal graphs. We give a polynomial time algorithm for bipartite graphs when the operation is edge contraction, the parameter is the independence number and d is fixed. Further, we complete the complexity dichotomy for H-free graphs when the parameter is the clique number and the operation is edge contraction by showing that this problem is NP-hard in (C3+P1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_3+P_1)$$\\end{document}-free graphs even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}. When the operation is edge deletion and the parameter is the chromatic number, we determine the computational complexity of the associated problem for cographs and complete multipartite graphs. Our results answer several open questions stated in Diner et al. (Theor Comput Sci 746:49–72, 2012, https://doi.org/10.1016/j.tcs.2018.06.023).

Crossing Numbers of Join Product with Discrete Graphs: A Study on 6-Vertex Graphs

Reducing the number of crossings on graph edges can be useful in various applications, including network visualization, circuit design, graph theory, cartography or social choice theory. This paper aims to determine the crossing number of the join product G*+Dn, where G* is a connected graph isomorphic to K2,2,2∖{e1,e2} obtained by removing two edges e1,e2 with a common vertex and a second vertex from the different partitions of the complete tripartite graph K2,2,2, and Dn is a discrete graph composed of n isolated vertices. The proofs utilize known exact crossing number values for join products of specific subgraphs Hk of G* with discrete graphs in combination with the separating cycles. Similar approaches can potentially estimate unknown crossing numbers of other six-vertex graphs with a larger number of edges in join products with discrete graphs, paths or cycles.

Regular and Maximal Graphs with Prescribed Tripartite Graph as a Star Complement

Regular and Maximal Graphs with Prescribed Tripartite Graph as a Star Complement

Approximation Algorithms for Partial Vertex Covers in Trees

This paper is concerned with designing algorithms for and analyzing the computational complexity of the partial vertex cover problem in trees. Graphs (and trees) are frequently used to model risk management in various systems. In particular, Caskurlu et al. in [4] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. In this paper, we focus on a special case of the partial vertex cover problem, when the input graph is a tree. We consider four possible versions of this setting, depending on whether or not, the vertices and edges are weighted. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable. However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, remained open. The main contribution of this paper is to resolve these questions by fully characterizing which variants of partial vertex cover remain NP-hard in trees, and which can be solved in polynomial time. In the paper, we propose two pseudo-polynomial DP-based algorithms for the most general case in which weights are present on both the edges and the vertices of the tree. One of these algorithms leads to a polynomial-time procedure, when weights are confined to the edges of the tree. The insights used in this algorithm are combined with additional scaling ideas to derive an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.

Combinatorial explanation of the weighted Laplacian characteristic polynomial of a graph and applications

Assume that G is a graph with n vertices v1,v2,…,vn. Given a rooted spanning forest F of G, which has s components and roots vk1,vk2,…,vks, define the weight ω(F) of F to be ∏i=1sxki, where x1,x2,…,xn are n independent commuting variables. In this paper, we give combinatorial explanations of the weighted enumeration of rooted spanning forests of G, the Laplacian characteristic polynomial of G, and the weighted Laplacian det[L+diag(x1,x2,…,xn)], where L is the Laplacian matrix of G and diag(x1,x2,…,xn) is a diagonal matrix. As applications, we count rooted spanning forests in the almost complete bipartite graph, and we also give a new proof of a formula on the number of rooted spanning forests in a complete multipartite graph Ka1,a2,…,as each of which has bi roots in the i-th part for i=1,2,…,s.

Computing the anti-Ramsey number for trees in complete tripartite graph

The anti-Ramsey number AR(K,F) is the largest color number in an edge-coloring of K that contains no rainbow copies of graphs in the family F. In this paper, we determine the exact value of the anti-Ramsey number for trees in a balanced complete tripartite graph.

Extremal numbers of disjoint triangles in r-partite graphs

For two graphs G and F, the extremal number of F in G, denoted by ex(G,F), is the maximum number of edges in a spanning subgraph of G not containing F as a subgraph. Determining ex(Kn,F) for a given graph F is a classical extremal problem in graph theory. In 1962, Erdős determined ex(Kn,kK3), which generalized Mantel's Theorem. On the other hand, in 1974, Bollobás, Erdős, and Straus determined ex(Kn1,n2,…,nr,Kt), which extended Turán's Theorem to complete multipartite graphs. In this paper, we determine ex(Kn1,n2,…,nr,kK3) for r≥4 and 10k−4≤n1+4k≤n2≤n3≤⋯≤nr.

Modeling User Reviews through Bayesian Graph Attention Networks for Recommendation

Recommender systems relieve users from cognitive overloading by predicting preferred items for users. Due to the complexity of interactions between users and items, graph neural networks (GNN) use graph structures to effectively model user–item interactions. However, existing GNN approaches have the following limitations: (1) User reviews are not adequately modeled in graphs. Therefore, user preferences and item properties that are described in user reviews are lost for modeling users and items; and (2) GNNs assume deterministic relations between users and items, which lack the stochastic modeling to estimate the uncertainties in neighbor relations. To mitigate the limitations, we build tripartite graphs to model user reviews as nodes that connect with users and items. We estimate neighbor relations with stochastic variables and propose a Bayesian graph attention network (i.e., ContGraph) to accurately predict user ratings. ContGraph incorporates the prior knowledge of user preferences to regularize the posterior inference of attention weights. Our experimental results show that ContGraph significantly outperforms 13 state-of-the-art models and improves the best performing baseline (i.e., ANR) by 5.23% on 25 datasets in the five-core version. Moreover, we show that correctly modeling the semantics of user reviews in graphs can help express the semantics of users and items.

Towards associating negative experiences and recommendations reported by Hajj pilgrims in a mass-scale survey

The annual Hajj presents diversified negative experiences to millions of pilgrims worldwide. The negative experiences and recommendations to overcome them as per pilgrims' feedback are yet to be analyzed from an aggregated perspective in the literature, which we do in this paper. To do so, first, we perform a large-scale survey (n=988) using our comprehensive questionnaire. Then, we perform both quantitative (e.g., clustering) and qualitative (e.g., thematic) analyses on the survey data. Our quantitative analysis reveals up to seven clusters of negative experiences. Further, going beyond the quantitative analysis, our qualitative analysis reveals 21 types of negative experiences, 20 types of recommendations, and nine themes connecting the negative experiences and recommendations. Accordingly, we reveal associations among the negative experiences and recommendations based on the themes in thematic analysis and present the associations through a tripartite graph. However, we have some limitations in this study, such as fewer female and young participants. In future, we plan to collect more responses from female and young participants and extend our work by analyzing linkages in the tripartite graph by augmenting the edges within the graph with appropriate weights. Overall, the findings of this study are expected to facilitate the prioritization of tasks for the management personnel in charge of the Hajj pilgrimage.