Complete Multipartite Research Articles

A Note on Decomposition of Tensor Product of Complete Multipartite Graphs into Gregarious Kite

A kite K is a graph which can be obtained by joining an edge to any vertex of K 3 . A kite with edge set { a b , b c , c a , c d } can be denoted as ( a , b , c ; c d ) . If every vertex of a kite in the decomposition lies in different partite sets, then we say that a kite decomposition of a multipartite graph is a gregarious kite decomposition. In this manuscript, it is shown that there exists a decomposition of ( K m ⊗ ¯¯¯¯¯ K n ) × ( K r ⊗ ¯¯¯¯¯ K s ) into gregarious kites if and only if n 2 s 2 m ( m − 1 ) r ( r − 1 ) ≡ 0 ( mod 8 ) , where ⊗ and × denote the wreath product and tensor product of graphs respectively. We denote a gregarious kite decomposition as G K -decomposition.

Retracted: Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

Retracted: Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

Complete Multipartite Graphs Decompositions Using Mutually Orthogonal Graph Squares

Complete Multipartite Graphs Decompositions Using Mutually Orthogonal Graph Squares

Eigenvalues of Graph Laplacians Via Rank-One Perturbations

Abstract We show how the spectrum of a graph Laplacian changes with respect to a certain type of rank-one perturbation. We apply our finding to give new short proofs of the spectral version of Kirchhoff’s Matrix Tree Theorem and known derivations for the characteristic polynomials of the Laplacians for several well-known families of graphs, including complete, complete multipartite, crown and threshold graphs.

On complete multipartite derangement graphs

Given a finite transitive permutation group G ≤ Sym ( Ω ) , with | Ω | ≥ 2 , the derangement graph Γ G of G is the Cayley graph Cay ( G , Der ( G )) , where Der ( G ) is the set of all derangements of G . Meagher et al. [On triangles in derangement graphs, J. Combin. Theory Ser. A , 180:105390, 2021] recently proved that Sym (2) acting on {1, 2} is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite. This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we prove that if p is an odd prime and G is a transitive group of degree 2 p , then the independence number of Γ G is at most twice the size of a point-stabilizer of G .

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

In this paper we consider a problem of job scheduling on parallel machines with a presence of incompatibilities between jobs. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine. We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). We consider a variety of machine types in our paper: identical, uniform, and unrelated. Our results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. We also provide algorithms, either polynomial exact algorithms for the easier problems, or algorithms with a guaranteed constant worst-case approximation ratio. In particular, we fill the gap on research for the problem of finding a schedule with the smallest ∑Cj on uniform machines. We address this problem by developing a linear programming relaxation technique with an appropriate rounding, which to our knowledge is a novelty for this criterion in the considered setting.

Optimal Wirelength of Balanced Complete Multipartite Graphs onto Cartesian Product of {Path, Cycle} and Trees

In any interconnection network, task allocation plays a major role in the processor speed as fair distribution leads to enhanced performance. Complete multipartite networks serve well for this purpose as the task can be split into different partites which improves the degree of reliability of the network. Such an allocation process in the network can be done by means of graph embedding. The optimal wirelength of a graph embedding helps in the distribution of deterministic algorithms from the guest graph to other host graphs in order to incorporate its unique deterministic properties on that chosen graph. In this paper, we propose an algorithm to compute the optimal wirelength of balanced complete multipartite graphs onto the Cartesian product of trees with path and cycle. Moreover, we derive the closed formulae for wirelengths in specific trees like (1-rooted) complete binary tree and sibling graphs.

The List-Chromatic Number of Complete Multipartite Hypergraphs and Multiple Covers by Independent Sets

This paper deals with list colorings of uniform hypergraphs. Let H(m, r, k) be the complete r-partite k-uniform hypergraph with parts of equal size m in which each edge contains exactly one vertex from some k ≤ r parts. Using results on multiple covers by independent sets, we establish that, for fixed k and r, the list-chromatic number of H(m, r, k) is (1 + o(1)) logr/(r−k+1)(m) as m → ∞.

On the additive chromatic number of several families of graphs

The Additive Coloring Problem is a variation of the Coloring Problem where labels of {1,…,k} are assigned to the vertices of a graph G so that the sum of labels over the neighborhood of each vertex is a proper coloring of G. The least value k for which G admits such labeling is called additive chromatic number of G. This problem was first presented by Czerwiński, Grytczuk and Żelazny who also proposed a conjecture that for every graph G, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete graphs, trees, non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. In this work, we show that the conjecture holds for split graphs. We also present exact formulas for computing the additive chromatic number for some subfamilies of split graphs (complete split, headless spiders and complete sun), regular bipartite, complete multipartite, fan, windmill, circuit, wheel, cycle sun and wheel sun.

Addressing Johnson Graphs, Complete Multipartite Graphs, Odd Cycles, and Random Graphs

Graham and Pollak showed that the vertices of any graph G can be addressed with N-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length N is minimum possible. In this article, we determine an addressing of length for the Johnson graphs J(n, k) and we show that our addressing is optimal when k = 1 or when , but not when n = 6 and k = 3. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to 10 vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on n vertices have an addressing of length at most .

Complete Multipartite Graphs Are Determined by Their Distance Spectra: A Short Proof

Lin et al. (2013) [16] conjectured that, the complete multipartite graphs are determined by their distance spectra. This was confirmed by Jin and Zhang (2014) [14]. In the present paper we give a shorter and more readable proof of the conjecture.

Connectivity and Hamiltonicity of Canonical Colouring Graphs of Bipartite and Complete Multipartite Graphs

A k-colouring of a graph G with colours 1 , 2 , … , k is canonical with respect to an ordering π = v 1 , v 2 , … , v n of the vertices of G if adjacent vertices are assigned different colours and, for 1 ≤ c ≤ k , whenever colour c is assigned to a vertex v i , each colour less than c has been assigned to a vertex that precedes v i in π . The canonical k-colouring graph of G with respect to π is the graph Can k π ( G ) with vertex set equal to the set of canonical k-colourings of G with respect to π , with two of these being adjacent if and only if they differ in the colour assigned to exactly one vertex. Connectivity and Hamiltonicity of canonical colouring graphs of bipartite and complete multipartite graphs is studied. It is shown that for complete multipartite graphs, and bipartite graphs there exists a vertex ordering π such that Can k π ( G ) is connected for large enough values of k. It is proved that a canonical colouring graph of a complete multipartite graph usually does not have a Hamilton cycle, and that there exists a vertex ordering π such that Can k π ( K m , n ) has a Hamilton path for all k ≥ 3 . The paper concludes with a detailed consideration of Can k π ( K 2 , 2 , … , 2 ) . For each k ≥ χ and all vertex orderings π , it is proved that Can k π ( K 2 , 2 , … , 2 ) is either disconnected or isomorphic to a particular tree.

On Locally-Balanced 2-Partitions of Complete Multipartite Graphs

A 2-partition of a graph G is a function f : V (G) →{White; Black}. A 2-partition f of a graph G is locally-balanced with an open neighborhood if for every ϑ ϵ V (G), ||{u ϵ NG(v): f (u) = White} |-| {u ϵ NG(v): f(u) = Black} ≤1; where NG(v) ={u ϵ V (G): uv ϵ E(G) }. A 2-partition f՛ 0 of a graph G is locallybalanced with a closed neighborhood if for every v ϵ V(G), ||{u ϵ NG(v): f՛ (u) = White} |-| {u ϵ NG[v]:; where NG[v] = NG(v)∪{v.} In this paper we give necessary and su±cient conditions for the existence of locally-balanced 2-partitions of complete multipartite graphs.

Fair 1-Factorizations and Fair Holey 1-Factorizations of Complete Multipartite Graphs

A k-factor of a graph G is a k-regular spanning subgraph of G. A k-factorization is a partition of E(G) into k-factors. Let K(n, p) be the complete multipartite graph with p parts, each of size n. If $$V_{1},\ldots , V_{p}$$V1,?,Vp are the p parts of V(K(n, p)), then a holeyk-factor of deficiency $$V_{i}$$Vi of K(n, p) is a k-factor of $$K(n,p)-V_{i}$$K(n,p)-Vi for some i satisfying $$1\le i \le p$$1≤i≤p. Hence a holeyk-factorization is a set of holey k-factors whose edges partition E(K(n, p)). Representing each (holey) k-factor as a color class in an edge-coloring, a (holey) k-factorization of K(n, p) is said to be fair if between each pair of parts the color classes have size within one of each other (so the edges are shared "evenly" among the permitted (holey) factors). In this paper the existence of fair 1-factorizations of K(n, p) is completely settled, as is the existence of fair holey 1-factorizations of K(n, p). The latter result can be used to provide a new construction for symmetric quasigroups of order np with holes of size n. Such quasigroups have the additional property that the permitted symbols are shared as evenly as possible among the cells in each $$n \times n$$n×n "box". These quasigroups are in some sense as far from frames produced by direct products as possible.

Complete Multipartite Graphs and their Null Set

For every natural number h, a graph G is said to be h-magic if there exists a labelling l:E(G)→Zh\\{0} such that the induced vertex set labelling l+:V(G)→Zh defined byl+(v)=∑uv∈E(G)l(uv), is a constant map. When this constant is zero, it is said that G admits a zero-sum h-magic labelling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h∈N such that G admits an h-zero-sum magic labelling. In 2007, E. Salehi determined the null set of complete bipartite graphs. In this paper we generalize this result by obtaining the null set of complete multipartite graphs.

Choosability with Separation of Complete Multipartite Graphs and Hypergraphs

AbstractFor a hypergraph G and a positive integer s, let be the minimum value of l such that G is L‐colorable from every list L with for each and for all . This parameter was studied by Kratochvíl, Tuza, and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of for balanced complete multipartite graphs and for complete k‐partite k‐uniform hypergraphs.

A note on sparseness conditions on chordless vertices of cycles

Conditions that disperse the chordless vertices of cycles correspond—sometimes in unexpected ways—to a variety of standard graph classes. Two examples: no cycle of length at least 5 has chordless vertices that are distance 2 apart in the cycle if and only if the graph is distance-hereditary; no cycle of length at least 5 has any chordless vertices if and only if every block of the graph is complete multipartite.

On the Fon-Der-Flaass interpretation of extremal examples for Turán’s (3, 4)-problem

Fon-Der-Flaass (1988) presented a general construction that converts an arbitrary \(\vec C_4 \)-free oriented graph Γ into a Turan (3, 4)-graph. He observed that all Turan-Brown-Kostochka examples result from his construction, and proved the lower bound \(\tfrac{4} {9} \) (1 − o(1)) on the edge density of any Turan (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound \(\tfrac{3} {7} \) (1 − o(1)) on the edge density of any Turan (3, 4)-graph resulting from the Fon-Der-Flaass construction under any of the following assumptions on the undirected graph G underlying the oriented graph Γ: (i) G is complete multipartite; (ii) the edge density of G is not less than \(\tfrac{2} {3} - \varepsilon \) for some absolute constant e > 0. We are also able to improve Fon-Der-Flaass’s bound to \(\tfrac{7} {{16}} \) (1 − o(1)) without any extra assumptions on Γ.

Vertex partition of a complete multipartite graph into two kinds of induced subgraphs

Let k , ℓ be positive integers. Let G be a graph of order k ℓ . It is shown that if G is a complete multipartite graph, G has a vertex partition V ( G ) = V 1 ∪ V 2 ∪ ⋯ ∪ V ℓ such that for some pair of graphs H 1 and H 2 of order k , the subgraph of G induced by V i is isomorphic to H 1 or H 2 for any i with 1 ≤ i ≤ ℓ . Furthermore, for graphs not necessarily complete multipartite, similar problems are discussed.

On graphs with complete multipartite [formula omitted]-graphs

Jurišić and Koolen proposed to study 1-homogeneous distance-regular graphs, whose μ -graphs (that is, the graphs induced on the common neighbours of two vertices at distance 2) are complete multipartite. Examples include the Johnson graph J (8, 4), the halved 8-cube, the known generalized quadrangle of order (4, 2), an antipodal distance-regular graph constructed by T. Meixner and the Patterson graph. We investigate a more general situation, namely, requiring the graphs to have complete multipartite μ -graphs, and that the intersection number α exists, which means that for a triple ( x , y , z ) of vertices in Γ , such that x and y are adjacent and z is at distance 2 from x and y , the number α ( x , y , z ) of common neighbours of x , y and z does not depend on the choice of a triple. The latter condition is satisfied by any 1-homogeneous graph. Let K t × n denote the complete multipartite graph with t parts, each of which consists of an n -coclique. We show that if Γ is a graph whose μ -graphs are all isomorphic to K t × n and whose intersection number α exists, then α = t , as conjectured by Jurišić and Koolen, provided α ≥ 2 . We also prove t ≤ 4 , and that equality holds only when Γ is the unique distance-regular graph 3 . O 7 ( 3 ) .