Multipartite Graphs Research Articles

On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs

Graphs of order n with fault-tolerant metric dimension n have recently been characterized.This paper points out an error in the proof of this characterization. We show that the complete multipartite graphs also have the fault-tolerant metric dimension n, which provides an infinite family of counterexamples to the characterization. Furthermore, we find exact values of the metric, edge metric, mixed-metric dimensions, the domination number, locating-dominating number, and metric-locating-dominating number for the complete multipartite graphs. These results generalize various results in the literature from complete bipartite to complete multipartite graphs.

On infinite circulant-balanced complete multipartite graphs decompositions based on generalized algorithmic approaches

Graph theory is a powerful and essential tool for applied scientists and engineers in analyzing and designing algorithms for several problems. Graph theory has a vital role in complex systems, especially in computer sciences. Applications of graph theory can be found in many scientific disciplines such as operational research, engineering, life sciences, management sciences, coding, and computer science. In the literature, there are two algorithms for constructing decompositions for the circulant graphs C2r,r with 2r vertices and r degree. For r,m⩾2, a circulant-balanced complete multipartite graph Cmr,(m-1)r is a generalization to C2r,r. Here, the major contributions are the generalization of the algorithms used for constructing the decompositions for the circulant graphs C2r,r to generalized algorithms that can be used for the decomposition of Cmr,(m-1)r that have mr vertices and (m-1)r degree. These generalized algorithms manage us to decompose the circulant graphs Cmr,(m-1)r by different graph classes. The paper's novelty is demonstrated by the fact that it is the first to suggest general approaches for decomposing circulant-balanced complete multipartite graphs. The constructed results in the present paper are suitable for generating different codes and also have great importance in the design of experiments.

The size multipartite Ramsey numbers mj(C3,C3,nK2,mK2)

Assume that [Formula: see text] is a complete, and multipartite graph consisting of j partite sets and n vertices in each partite set. For given graphs [Formula: see text], the multipartite Ramsey number (M-R-number) [Formula: see text], is the smallest integer [Formula: see text], such that for any [Formula: see text]-edge-coloring [Formula: see text] of the edges of [Formula: see text], [Formula: see text] contains a monochromatic copy of [Formula: see text] for at least one [Formula: see text]. The size of M-R-number [Formula: see text] for [Formula: see text], the size of M-R-number [Formula: see text] for [Formula: see text] and [Formula: see text], and the size of M-R-number [Formula: see text] for [Formula: see text] and [Formula: see text] have been computed in several papers up to now. In this paper, we determine some lower bounds for the M-R-number [Formula: see text] for each [Formula: see text], and some values of M-R-number [Formula: see text] for some [Formula: see text], and each [Formula: see text].

Topology of clique complexes of line graphs

The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.

On the Unimodality of Domination Polynomials

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph G is the generating function of the number of dominating sets of each cardinality in G, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode $$ \lceil \frac{n}{2}\rceil $$ .

The Equidistant Dimension of Graphs

A subset S of vertices of a connected graph G is a distance-equalizer set if for every two distinct vertices x, y in V (G) {setminus } S there is a vertex w in S such that the distances from x and y to w are the same. The equidistant dimension of G is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related to 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.

Non-zero sum Heffter arrays and their applications

In this paper we introduce a new class of partially filled arrays that, as Heffter arrays, are related to difference families, graph decompositions and biembeddings. A non-zero sum Heffter arrayNH(m,n;h,k) is an m×n p.f. array with entries in Z2nk+1 such that: each row contains h filled cells and each column contains k filled cells; for every x∈Z2nk+1∖{0}, either x or −x appears in the array; the sum of the elements in every row and column is different from 0 (in Z2nk+1). Here first we explain the connections with relative difference families and with path decompositions of the complete multipartite graph. Then we present a complete solution for the existence problem and a constructive complete solution for the square case and for the rectangular case with no empty cells when the additional, very restrictive, property of being “globally simple” is required. Finally, we show how these arrays can be used to construct biembeddings of complete graphs.

Competitively orientable complete multipartite graphs

We say that a digraph D is competitive if any pair of vertices has a common out-neighbor in D and that a graph G is competitively orientable if there exists a competitive orientation of G. The competition graph of a digraph D is defined as the graph with the vertex set V(D) and an edge uv if and only if u and v compete in D. The notion of competitive digraph arose while studying digraph whose competition graphs are complete. We derive some useful properties of competitively orientable graphs and show that a complete graph of order n is competitively orientable if and only if n≥7. Then we completely characterize a competitively orientable complete multipartite graph in terms of the sizes of its partite sets. Moreover, we present a way to build a competitive multipartite tournament in each of competitively orientable cases.

A note on marginal entropy of graphs

In 2006, Konstantinova proposed the marginal entropy of a graph based on the Wiener index. In this paper, we obtain the marginal entropy of the complete multipartite graphs, firefly graphs, lollipop graphs, clique-chain graphs, Cartesian product and join of two graphs, which extends the results of Sahin.

Chasing a Drunk Robber in Many Classes of Graphs

A cops and robbers game (CR) on graphs was originated in 1983 by Quilliot and by Nowakowski and Winkler independently. This game has been applied in many problems in the area of theoretical computer science such as information seeking, robot motion planning or network security as evidenced by many conferences and publications. The CR game has also been extensively studied and varied to many versions. In this paper, we focus on CR game under the condition that the robber performs a random walk. Namely, he drunkenly, or randomly, chooses his next move to escape the cop, while the cop plays optimally in order to minimize the expected drunk capture times{textit{dct}}(G) of a graph G. We apply the concepts of expected hitting time in Markov Chain and combinatorial technique to find {textit{dct}}(G) for graphs G that have been used in many applications which are cycles, complete multipartite graphs, grid graphs and prism graphs.

Perfect state transfer in Grover walks between states associated to vertices of a graph

We study perfect state transfer in Grover walks, which are typical discrete-time quantum walk models. In particular, we focus on states associated to vertices of a graph. We call such states vertex type states. Perfect state transfer between vertex type states can be studied via Chebyshev polynomials. We derive a necessary condition on eigenvalues of a graph for perfect state transfer between vertex type states to occur. In addition, we perfectly determine the complete multipartite graphs whose partite sets are the same size on which perfect state transfer occurs between vertex type states, together with the time.

Approaches that output infinitely many graphs with small local antimagic chromatic number

An edge labeling of a connected graph [Formula: see text] is said to be local antimagic if it is a bijection [Formula: see text] such that for any pair of adjacent vertices x and y, [Formula: see text], where the induced vertex label [Formula: see text], with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by [Formula: see text], is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we show the existence of infinitely many bipartite and tripartite graphs with [Formula: see text].

Maximal sets of Hamilton cycles in complete multipartite graphs IV

Finding the values of μ for which there exists a maximal set of μ edge-disjoint Hamilton cycles in the complete multipartite graph Knp has been considered in papers for over 20 years. This paper finally settles the problem by finding such a set in the last remaining open case, namely where μ is as small as possible (so its existence was still in doubt) when n=3 and the number of parts, p, is 3 (mod 4).

Scheduling with complete multipartite incompatibility graph on parallel machines: Complexity and algorithms

In this paper, the problem of scheduling on parallel machines with a presence of incompatibilities between jobs is considered. 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.The paper provides several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). The problems are considered for a variety of machine types: identical, uniform and unrelated. The results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. Also, polynomial-time exact algorithms are provided for easier problems and algorithms with a guaranteed constant worst-case approximation ratio for harder ones.In particular, there is provided a polynomial-time approximation scheme (PTAS) for scheduling with respect to ∑Cj on uniform machines, when the number of parts (i.e. sets of vertices constituting the mentioned graph) is bounded. The problem is addressed by developing a linear programming relaxation technique with appropriate rounding. This technique together with an exhaustive search (albeit in a manner controlled by the precision parameter) allows to provide the desired algorithm. For Cmax a PTAS is provided for the case of unit time jobs, but when the number of parts is part of the input. Interestingly, the latter result shows a connection between the considered problem of scheduling and covering problems.

Counting spanning trees in almost complete multipartite graphs

Counting spanning trees in almost complete multipartite graphs

Toppleable permutations, excedances and acyclic orientations

Recall that an excedance of a permutation $\pi$ is any position $i$ such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on $n$ letters is the same as those for which excedances happen exactly at $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$. We also give a formula for the number of AUSOs of complete multipartite graphs. We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.Mathematics Subject Classifications: 05A19, 05A05, 05C30

ON INTERVAL EDGE-COLORINGS OF COMPLETE MULTIPARTITE GRAPHS

A graph $G$ is called a complete $r$-partite $(r\geq 2)$ graph, if its vertices can be divided into $r$ non-empty independent sets $V_1,\ldots,V_r$ in a way that each vertex in $V_i$ is adjacent to all the other vertices in $V_j$ for $1\leq i<j\leq r$. Let $K_{n_{1},n_{2},\ldots,n_{r}}$ denote a complete $r$-partite graph with independent sets $V_1,V_2,\ldots,V_r$ of sizes $n_{1},n_{2},\ldots,n_{r}$. An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is called an \emph{interval $t$-coloring}, if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. In this paper we have obtained some results on the existence and construction of interval edge-colorings of complete $r$-partite graphs. Moreover, we have also derived an upper bound on the number of colors in interval colorings of complete multipartite graphs.

Diagonal Ramsey numbers in multipartite graphs related to stars

Let the star on n vertices, namely K 1, n − 1 be denoted by S n . If every two coloring of the edges of a complete balanced multipartite graph K j × s there is a copy of S n in the first color or a copy of S m in the second color, then we will say K j × s → ( S n , S m ) . The size Ramsey multipartite number m j ( S n , S m ) is the smallest natural number s such that K j × s → ( S n , S m ) . In this paper, we obtain the exact values of the size Ramsey numbers m j ( S n , S m ) for n , m ≥ 3 and j ≥ 3 .

Indicated coloring of the Mycielskian of some families of graphs

Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round, the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben’s strategy) is called the indicated chromatic number of G, denoted by [Formula: see text]. In this paper, we examine whether the Mycielskian of G, [Formula: see text], is k-indicated colorable for all [Formula: see text], whenever G is l-indicated colorable for all [Formula: see text]. In this direction, we prove that the Mycielskian of the bipartite graphs, complete multipartite graphs, [Formula: see text]-free graphs, [Formula: see text]-free graphs, [Formula: see text]-free graphs and [Formula: see text]-free graphs are k-indicated colorable for all k greater than or equal to the indicated chromatic number of the corresponding graphs.

On maximum intersecting sets in direct and wreath product of groups

For a permutation group G acting on a set V, a subset I of G is said to be an intersecting set if for every pair of elements g,h∈I there exists v∈V such that g(v)=h(v). The intersection densityρ(G) of a transitive permutation group G is the maximum value of the quotient |I|/|Gv| where Gv is a stabilizer of a point v∈V and I runs over all intersecting sets in G. If Gv is the largest intersecting set in G then G is said to have the Erdős–Ko–Rado (EKR)-property, and moreover, G has the strict-EKR-property if every intersecting set of maximum size in G is a coset of a point stabilizer. Intersecting sets in G coincide with independent sets in the so-called derangement graphΓG, defined as the Cayley graph on G with connection set consisting of all derangements, that is, fixed-point free elements of G. In this paper a conjecture regarding the existence of transitive permutation groups whose derangement graphs are complete multipartite graphs, posed by Meagher, Razafimahatratra and Spiga in Meagher et al. (2021) is proved. The proof uses direct product of groups. Questions regarding maximum intersecting sets in direct and wreath products of groups and the (strict)-EKR-property of these group products are also investigated. In addition, some errors appearing in the literature on this topic are corrected.