M-partite Graph Research Articles

On Nordhaus-Gaddum type relations of δ-complement graphs

The δ-complement graphs were introduced by Amrithalakshmi et al. in 2022. In their work, some interesting properties of the graphs such as δ-self-complementary, adjacency, and hamiltonicity were studied. In this work, we study the coloring aspect of the δ-complement graphs. In particular, we provide lower and upper bounds on the product and the summation between the chromatic number and the δ-chromatic number of a graph, in the same fashion as the well-known Nordhaus-Gaddum type relations. Classes of graphs that achieve those bounds are also given. Furthermore, we provide upper bounds on δ-chromatic numbers in terms of the clique numbers and compute the δ-chromatic numbers of certain graphs including ladder graphs, path graphs, complete m-partite graphs, and small-world Farey graphs.

On Recognizing Staircase Compatibility

For the problem to find an m-clique in an m-partite graph, staircase compatibility has recently been introduced as a polynomial-time solvable special case. It is a property of a graph together with an m-partition of the vertex set and total orders on each subset of the partition. In optimization problems involving m-cliques in m-partite graphs as a subproblem, it allows for totally unimodular linear programming formulations, which have shown to efficiently solve problems from different applications. In this work, we address questions concerning the recognizability of this property in the case where the m-partition of the graph is given, but suitable total orders are to be determined. While finding these total orders is NP-hard in general, we give several conditions under which it can be done in polynomial time. For bipartite graphs, we present a polynomial-time algorithm to recognize staircase compatibility and show that staircase total orders are unique up to a small set of reordering operations. On m-partite graphs, where the recognition problem is NP-complete in the general case, we identify a polynomially solvable subcase and also provide a corresponding algorithm to compute the total orders. Finally, we evaluate the performance of our ordering algorithm for m-partite graphs on a set of artificial instances as well as real-world instances from a railway timetabling application. It turns out that applying the ordering algorithm to the real-world instances and subsequently solving the problem via the aforementioned totally unimodular reformulations indeed outperforms a generic formulation which does not exploit staircase compatibility.

Decomposition of product graphs into paths and stars on five vertices

Let Sk and Kk respectively denote a path, a star and a complete graph on k vertices. By a -decomposition of a graph G, we mean a decomposition of G into r copies of and s copies of In this paper, it shown that the graph admits a -decomposition if and only if where denotes a tensor product of complete graphs. Also we extend the existence of such a decomposition in complete m-partite graphs.

Continuous Monotonic Decomposition of Some Standard Graphs by using an Algorithm

In this paper we elaborate an algorithm to compute the necessary and sufficient conditions for the continuous monotonic star decomposition of the bipartite graph Km,r and the number of vertices and the number of disjoint sets. Also an algorithm to find the tensor product of Pn  Ps has continuous monotonic path decomposition. Finally we conclude that in this paper the results described above are complete bipartite graphs that accept Continuous monotonic star decomposition. There are many other classes of complete tripartite graphs that accept Continuous monotonic star decomposition. In this research article Extended to complete m-partite graphs for grater values of m. Also the algorithm can be developed for the tensor product of different classes such as Cn Wn K1,n , , with Pn

Multi-sided assignment games on m-partite graphs

We consider a multi-sided assignment game with the following characteristics: (a) the agents are organized in m sectors that are connected by a graph that induces a weighted m-partite graph on the set of agents, (b) a basic coalition is formed by agents from different connected sectors, and (c) the worth of a basic coalition is the addition of the weights of all its pairs that belong to connected sectors. We provide a sufficient condition on the weights to guarantee balancedness of the related multi-sided assignment game. Moreover, when the graph on the sectors is cycle-free, we prove the game is strongly balanced and the core is fully described by means of the cores of the underlying two-sided assignment games associated with the edges of this graph. As a consequence, the complexity of the computation of an optimal matching is reduced and existence of optimal core allocations for each sector of the market is guaranteed.

Cycle frames and the Oberwolfach problem

For an integer λ,G(λ) denotes a graph G with uniform edge-multiplicity λ. Let J be a subset of positive integers. A 2-regular subgraph of m-partite graph Km⊗In containing vertices of all but one partite set is called partial 2-factor, where ⊗ denotes wreath product and In is an independent set on n vertices. If (Km⊗In)(λ) can be partitioned into edge-disjoint partial 2-factors such that each partial 2-factor consists of cycles of lengths from J, then we say that (Km⊗In)(λ)has a(J,λ)-cycle frame. The Oberwolfach problem OP(m1α1,m2α2,…,mtαt), raised by Ringel, asks the existence of a 2-factorization of Kn (when n is odd) or Kn−I (when n is even), in which each 2-factor consists of exactly αi cycles of length mi, i=1,2,…,t. In this paper, we show that there exists a ({4,6},λ)-cycle frame of (Km⊗In)(λ) if and only if λn≡0(mod2), m≥3, (n,m)∉{(1,3),(2t+1,2s)∣t≥0,s≥2}. Further we show that there exists a ({3,4},1)-cycle frame of Km⊗In if and only if m≥3 and n≡0(mod2). As a consequence, we solve OP(3a,4b), OP(3a,6b) and OP(5a,10b) with some restrictions on a,b∈N.

Bounded colorings of multipartite graphs and hypergraphs

Let c be an edge-coloring of the complete n-vertex graph Kn. The problem of finding properly colored and rainbow Hamilton cycles in c was initiated in 1976 by Bollobás and Erdős and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek et al. (2012). We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph G.We also study multipartite analogues of these questions. We give (up to a constant factor) optimal sufficient conditions for a coloring c of the complete balanced m-partite graph to contain a properly colored or rainbow copy of a given graph G with maximum degree Δ. Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes Δ≫m and Δ≪m. Our main tool is the framework of Lu and Székely for the space of random bijections, which we extend to product spaces.

Multi-Sided Assignment Games on M-Partite Graphs

We consider a multi-sided assignment game with the following characteristics: (a) the agents are organized in m sectors that are connected by a graph that induces a weighted m-partite graph on the set of agents, (b) a basic coalition is formed by agents from different connected sectors, and (c) the worth of a basic coalition is the addition of the weights of all its pairs that belong to connected sectors. We provide a sufficient condition on the weights to guarantee balancedness of the related multi-sided assignment game. Moreover, when the graph on the sectors is cycle-free, we prove the game is strongly balanced and the core is described by means of the cores of the underlying two-sided assignment games associated with the edges of this graph. Moreover, once selected a spanning tree of the cycle-free graph on the sectors, the equivalence between core and competitive equilibria is established.

On [formula omitted]-equitable chromatic threshold of Kronecker products of complete graphs

Let r and k be positive integers. A graph G is r-equitably k-colorable if its vertex set can be partitioned into k independent sets, any two of which differ in size by at most r. The r-equitable chromatic threshold of a graph G, denoted by χr=∗(G), is the minimum k such that G is r-equitably k′-colorable for all k′≥k. Let G×H denote the Kronecker product of graphs G and H. In this paper, we completely determine the exact value of χr=∗(Km×Kn) for general m,n and r. As a consequence, we show that for r≥2, if n≥1r−1(m+r)(m+2r−1) then Km×Kn and its spanning supergraph Km(n) have the same r-equitable colorability, and in particular χr=∗(Km×Kn)=χr=∗(Km(n)), where Km(n) is the complete m-partite graph with n vertices in each part.

Graph partitions for the multidimensional assignment problem

In this paper, we consider two decomposition schemes for the graph theoretical description of the axial Multidimensional Assignment Problem (MAP). The problem is defined as finding n disjoint cliques of size m with minimum total cost in Km×n, which is an m-partite graph with n elements per dimension. Even though the 2-dimensional assignment problem is solvable in polynomial time, extending the problem to include n≥3 dimensions renders it \(\mathcal{NP}\)-hard. We propose two novel decomposition schemes for partitioning a MAP into disjoint subproblems, that can then be recombined to provide both upper and lower bounds to the original problem. For each of the partitioning schemes, we investigate and compare the efficiency of distinct exact and heuristic methodologies, namely augmentation and partitioning. Computational results for the methods, along with a hybrid one that consists of both partitioning schemes, are presented to depict the success of our approaches on large-scale instances.

Cycle Frames of Complete Multipartite Multigraphs - III

For two graphs G and H their wreath product has vertex set in which two vertices and are adjacent whenever or and . Clearly, , where is an independent set on n vertices, is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices. A 2-regular subgraph of the complete multipartite graph containing vertices of all but one partite set is called partial 2-factor. For an integer λ, denotes a graph G with uniform edge multiplicity λ. Let J be a set of integers. If can be partitioned into edge-disjoint partial 2-factors consisting cycles of lengths from J, then we say that has a -cycle frame. In this paper, we show that for and , there exists a -cycle frame of if and only if and . In fact our results completely solve the existence of a -cycle frame of .

C 4p -frame of complete multipartite multigraphs

For two graphs G and H their wreath product\({G \otimes H}\) has the vertex set \({V(G) \times V(H)}\) in which two vertices (g1, h1) and (g2, h2) are adjacent whenever \({g_{1}g_{2} \in E(G)}\) or g1 = g2 and \({h_{1}h_{2} \in E(H)}\) . Clearly \({K_{m} \otimes I_{n}}\) , where In is an independent set on n vertices, is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices. A subgraph of the complete multipartite graph \({K_m \otimes I_n}\) containing vertices of all but one partite set is called partial factor. An H-frame of \({K_m \otimes I_n}\) is a decomposition of \({K_m \otimes I_n}\) into partial factors such that each component of it is isomorphic to H. In this paper, we investigate C2k-frames of \({(K_m \otimes I_n)(\lambda)}\) , and give some necessary or sufficient conditions for such a frame to exist. In particular, we give a complete solution for the existence of a C4p-frame of \({(K_m \otimes I_n)(\lambda)}\) , where p is a prime, as follows: For an integer m ≥ 3 and a prime p, there exists a C4p-frame of \({(K_m \otimes I_n)(\lambda)}\) if and only if \({(m-1)n \equiv 0 ({\rm {mod}} {4p})}\) and at least one of m, n must be even, when λ is odd.

More results on cycle frames and almost resolvable cycle systems

Let J be a set of positive integers. Suppose m>1 and H is a complete m-partite graph with vertex set V and m groups G1,G2,…,Gm. Let |V|=v and G={G1,G2,…,Gm}. If the edges of λH can be partitioned into a set C of cycles with lengths from J, then (V,G,C) is called a cycle group divisible design with index λ and order v, denoted by (J,λ)-CGDD. A (J,λ)-cycle frame is a (J,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V∖Gi into cycles for some Gi∈G. The existence of (k,λ)-cycle frames of type gu with 3≤k≤6 has been solved completely. In this paper, we show that there exists a ({3,5},λ)-cycle frame of type gu for any u≥4, λg≡0(mod2), (g,u)≠(1,5),(1,8) and (g,u,λ)≠(2,5,1). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). It has been proved that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three exceptions and four possible exceptions. In this paper, we shall show that there exists a k-ARCS(2kt+1) for all t≥1, 11≤k≤49, k≡1(mod2) and t≠2,3,5.

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph Km(n1,n2,…,nm) with vertex set V and m independent sets G1,G2,…,Gm of n1,n2,…,nm vertices respectively. Let G={G1,G2,…,Gm}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V∖Gi for some Gi∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type gu. In this paper, we show that there exists a (k,λ)-cycle frame of type gu for k∈{4,5,6} if and only if g(u−1)≡0(modk), λg≡0(mod2), u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for k≡0(mod2). In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when k≡1(mod2). We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.

K-Kernels in Multipartite Tournaments

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∊ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∊ V(D) – N then there exists v ∊ N such that d(u, v) ≤ l). A k-kernel is a (k, k – 1)-kernel. An m-partite tournament is an orientation of an m-partite complete graph. In this paper we use a tool for finding sufficient conditions for a digraph to have a k-kernel, the k-closure of a digraph, which is employed to prove that every m-partite tournament has a k-kernel for every k ≥ 4, m ≥ 2, and to give a simple proof of the fact that for k ≥ 2 it is NP-complete to decide if a digraph D has a k-kernel, or not. We also characterize the m-partite tournaments that have a 3-kernel for m ≥ 2 as those m-partite tournaments having a 2-absorbent vertex for the directed cycles of length four.

Packing 5-cycles into balanced complete m-partite graphs for odd m

Let \(K_{n_{1},n_{2},\ldots,n_{m}}\) be a complete m-partite graph with partite sets of sizes n 1,n 2,…,n m . A complete m-partite graph is balanced if each partite set has n vertices. We denote this complete m-partite graph by K m(n). In this paper, we completely solve the problem of finding a maximum packing of the balanced complete m-partite graph K m(n), m odd, with edge-disjoint 5-cycles and we explicitly give the minimum leaves.

Formula omitted]-decompositions of some regular graphs

In this paper, for any prime p ⩾ 11 , we consider C p -decompositions of K m × K n and K m * K ¯ n and also C p -factorizations of K m × K n , where × and * denote the tensor product and wreath product of graphs, respectively, ( K m * K ¯ n is isomorphic to the complete m-partite graph in which each partite set has exactly n vertices). It has been proved that for m , n ⩾ 3 , C p -decomposes K m × K n if and only if (1) either m or n is odd and (2) p | mn ( m - 1 ) ( n - 1 ) . Further, it is shown that for m ⩾ 3 , C p -decomposes K m * K ¯ n if and only if (1) ( m - 1 ) n is even and (2) p | m ( m - 1 ) n 2 . Except possibly for some valid pairs of integers m and n , the necessary conditions for the existence of C p -factorization of K m × K n are proved to be sufficient.

On integral graphs which belong to the class αKa,a,...,a,b,b,...,b

Let G be a simple graph and let G- denote its complement. We say that G is integral if its spectrum consists of integral values. Let Kxa, yb = Ka,a,...,a,b,b...,b be the complete m-partite graph with xa + yb vertices where x and y are positive integers and m = x + y. In this work we consider integral graphs which belong to the class ?Kxa,yb for any ? > 1 and a > b where mG denotes the m-fold union of the graph G.

Existence of cyclic k-cycle systems of the complete graph

Starting from earliest papers by Rosa we solve, directly and explicitly, the existence problem for cyclic k-cycle systems of the complete graph K v with v≡1 ( mod 2k) , and the existence problem for cyclic k-cycle systems of the complete m-partite graph K m× k with m and k being odd. As a particular consequence, a cyclic p-cycle system of K v with p being a prime exists for all admissible values of v but ( p, v)≠(3,9). This was previously known only for p=3,5,7.

Efficiency of applying Hopfield neural networks with simulated annealing and genetic algorithms for solving m-partite graph problem

An m-partite graph is defined as a graph that consists of m nodes each of which contains a set of elements, and the arcs connecting elements from different nodes. Each element in this graph comprises its specific attributes such as cost and resources. The weighted values of arcs represent the dissimilarities of resources between elements from different nodes. The m-partite graph problem is defined as selecting exactly one representative from a set of elements for each node in such a way that the sum of both the costs of the selected elements and their dissimilarities is minimised. In order to solve such a problem, Hopfield neural networks based approach is adopted in this paper. The Liapunov function (energy function) of Hopfield neural networks specially designed for solving m-partite graph problem is constructed. In order to prohibit Hopfield neural networks from becoming trapped in their local minima, simulated annealing and genetic algorithms are thus utilised and combined with Hopfield neural networks to get globally optimal solution to m-partite graph problem. The result of the approaches developed in this paper shows the definitive promise for leading to the optimal solution to the m-partite graph problem compared with that of other currently available algorithms.