Complete Bipartite Graph Research Articles

Some modified types of pitchfork domination and it's inverse

Let G be a finite, simple graph, without isolated vertices. For any non-negative integers x and y, a set D ⊆ V is a ”pitchfork dominating set pds”, when every vertex in D, dominates at most y and at least x vertices of V − D. A subset D−1 of V − D is an inverse pds if it is a pitchfork set. The pitchfork domination number of G, γpf (G), is the number of elements of a smallest pds. The ”inverse pitchfork domination number” of G, γ −1 pf (G), is the number of elements of a smallest inverse pds. In this paper, some modified pitchfork dominations and its inverse dominations are introduced when x = 1 and y = 2. Several bounds and properties are given and proved. Then, these modified dominations are applied on some standard graphs such as: path, cycle, wheel, complete, complete bipartite graph and their complements.

On the decomposition of circulant graphs using algorithmic approaches

Many structural models in chemistry, biology, computer science, sociology, and operations research can be analyzed using graph theory. Some examples of these structure models are species movement between regions, molecular bonds, shortest spanning trees, development of computer algorithms. This paper introduces the edge decomposition of circulant graphs with 2n vertices by different graph classes. These circulant graphs are denoted as C2n,n, where n is the degree of the graph C2n,n. We propose two algorithmic approaches for constructing edge decomposition of C2n,n. With aid of the proposed algorithmic approaches, we construct paths decompositions, trees decompositions, disjoint unions of cycles and trees decompositions, and complete bipartite graphs decompositions of C2n,n. In the end, a recursive construction of an edge decomposition, based on orthogonal edge decompositions, is proposed. This helps in using the mutually orthogonal decompositions of the circulant graphs C2n,n in several applications, such as the design of experiments and binary codes generation.

An average degree condition for independent transversals

In 1994, Erdős, Gyárfás and Łuczak posed the following problem: given disjoint vertex sets V1,…,Vn of size k, with exactly one edge between any pair Vi,Vj, how large can n be such that there will always be an independent transversal? They showed that the maximal n is at most (1+o(1))k2, by providing an explicit construction with these parameters and no independent transversal. They also proved a lower bound which is smaller by a 2e-factor.In this paper, we solve this problem by showing that their upper bound construction is best possible: if n≤(1−o(1))k2, there will always be an independent transversal. In fact, this result is a very special case of a much more general theorem which concerns independent transversals in arbitrary partite graphs that are ‘locally sparse’, meaning that the maximum degree between each pair of parts is relatively small. In this setting, Loh and Sudakov provided a global maximum degree condition for the existence of an independent transversal. We show that this can be relaxed to an average degree condition.We can also use our new theorem to establish tight bounds for a more general version of the Erdős–Gyárfás–Łuczak problem and solve a conjecture of Yuster from 1997. This exploits a connection to the Turán numbers of complete bipartite graphs, which might be of independent interest.

The Strong Equitable Vertex 1-Arboricity of Complete Bipartite Graphs and Balanced Complete k-Partite Graphs

An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable (q,r)-tree-coloring of a graph G is an equitable q-coloring of G such that the subgraph induced by each color class is a forest of maximum degree at most r. Let the strong equitable vertex r-arboricity of a graph G, denoted by var≡(G), be the minimum p such that G has an equitable (q,r)-tree-coloring for every q≥p. The values of va1≡(Kn,n) were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of va1≡(Kn,n) were found in some special cases. In this paper, we extend their results by giving the exact values of va1≡(Kn,n) for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each va1≡(Km,n) and va1≡(G) where G is a balanced complete k-partite graph Kn,…,n. Both complete bipartite graphs Km,n and balanced complete k-partite graphs Kn,…,n are symmetry in several aspects and also studied broadly. For the other aspect of symmetry, by the definition of equitable k-coloring of graphs, in a specific case that the number of colors divides the number of vertices of graph, we can say that the graph is a balanced k-partite graph.

Hat Guessing on Books and Windmills

The hat-guessing number is a graph invariant defined by Butler, Hajiaghayi, Kleinberg, and Leighton. We determine the hat-guessing number exactly for book graphs with sufficiently many pages, improving previously known lower bounds of He and Li and exactly matching an upper bound of Gadouleau. We prove that the hat-guessing number of $K_{3,3}$ is $3$, making this the first complete bipartite graph $K_{n,n}$ for which the hat-guessing number is known to be smaller than the upper bound of $n+1$ of Gadouleau and Georgiou. Finally, we determine the hat-guessing number of windmill graphs for most choices of parameters.

Nonlinear normal modes in a network with cubic couplings

<abstract><p>We consider a network with cubic couplings. This is related to the well known Fermi-Pasta-Ulam-Tsingou model. We show that nonlinear periodic orbits extend from particular eigenvectors of the graph Laplacian, these are termed <italic>nonlinear normal modes</italic>. We present large classes of graphs where this occurs. These are the graphs whose Laplacian eigenvectors have components in $ \{1, -1\} $ (bivalent), and $ \{1, -1, 0\} $ with a condition (soft-regular trivalent), the bipartite complete graphs and their extensions obtained by adding an edge between vertices having the same component. Finally, we study the stability of these solutions for chains and cycles.</p></abstract>

On Recent Advances in Divisor Cordial Labeling of Graphs

An assignment of intergers to the vertices of a graph <img src=image/13424638_01.gif> subject to certain constraints is called a vertex labeling of <img src=image/13424638_01.gif>. Different types of graph labeling techniques are used in the field of coding theory, cryptography, radar, missile guidance, <img src=image/13424638_02.gif>-ray crystallography etc. A DCL of <img src=image/13424638_01.gif> is a bijective function <img src=image/13424638_03.gif> from node set <img src=image/13424638_04.gif> of <img src=image/13424638_01.gif> to <img src=image/13424638_05.gif> such that for each edge <img src=image/13424638_06.gif>, we allot 1 if <img src=image/13424638_07.gif> divides <img src=image/13424638_08.gif> or <img src=image/13424638_08.gif> divides <img src=image/13424638_07.gif> & 0 otherwise, then the absolute difference between the number of edges having 1 & the number of edges having 0 do not exceed 1, i.e., <img src=image/13424638_09.gif>. If <img src=image/13424638_01.gif> permits a DCL, then it is called a DCG. A complete graph <img src=image/13424638_10.gif>, is a graph on <img src=image/13424638_11.gif> nodes in which any 2 nodes are adjacent and lilly graph <img src=image/13424638_12.gif> is formed by <img src=image/13424638_13.gif> joining <img src=image/13424638_14.gif>, <img src=image/13424638_15.gif> sharing a common node. i.e., <img src=image/13424638_16.gif>, where <img src=image/13424638_17.gif> is a complete bipartite graph & <img src=image/13424638_18.gif> is a path on <img src=image/13424638_11.gif> nodes. In this paper, we propose an interesting conjecture concerning DCL for a given <img src=image/13424638_01.gif>, besides, discussing certain general results concerning DCL of complete graph <img src=image/13424638_10.gif>-related graphs. We also prove that <img src=image/13424638_12.gif> admits a DCL for all <img src=image/13424638_15.gif>. Further, we establish the DCL of some <img src=image/13424638_12.gif>-related graphs in the context of some graph operations such as duplication of a node by an edge, node by a node, extension of a node by a node, switching of a node, degree splitting graph, & barycentric subdivision of the given <img src=image/13424638_01.gif>.

Rough sets in graphs using similarity relations

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.&lt;/p&gt;&lt;/abstract&gt;

Energy of Certain Classes of Graphs Determined by Their Laplacian Degree Product Adjacency Spectrum

In this study, we investigate the Laplacian degree product spectrum and corresponding energy of four families of graphs, namely, complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with a null graph.

Construction of Mutually Orthogonal Graph Squares Using Novel Product Techniques

Sets of mutually orthogonal Latin squares prescribe the order in which to apply different treatments in designing an experiment to permit effective statistical analysis of results, they encode the incidence structure of finite geometries, they encapsulate the structure of finite groups and more general algebraic objects known as quasigroups, and they produce optimal density error‐correcting codes. This paper gives some new results on mutually orthogonal graph squares. Mutually orthogonal graph squares generalize orthogonal Latin squares interestingly. Mutually orthogonal graph squares are an area of combinatorial design theory that has many applications in optical communications, wireless communications, cryptography, storage system design, algorithm design and analysis, and communication protocols, to mention just a few areas. In this paper, novel product techniques of mutually orthogonal graph squares are considered. Proposed product techniques are the half‐starters’ vectors Cartesian product, half‐starters’ function product, and tensor product of graphs. It is shown that by taking mutually orthogonal subgraphs of complete bipartite graphs, one can obtain enough mutually orthogonal subgraphs in some larger complete bipartite graphs. Also, we try to find the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques. As a direct application to the proposed different product techniques, mutually orthogonal graph squares for disjoint unions of stars are constructed. All the constructed results in this paper can be used to generate new graph‐orthogonal arrays and new authentication codes.

Ramsey properties of algebraic graphs and hypergraphs

Abstract One of the central questions in Ramsey theory asks how small the largest clique and independent set in a graph on N vertices can be. By the celebrated result of Erdős from 1947, a random graph on N vertices with edge probability $1/2$ contains no clique or independent set larger than $2\log _2 N$ , with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools. Say that an r-uniform hypergraph $\mathcal {H}$ is algebraic of complexity $(n,d,m)$ if the vertices of $\mathcal {H}$ are elements of $\mathbb {F}^{n}$ for some field $\mathbb {F}$ , and there exist m polynomials $f_1,\dots ,f_m:(\mathbb {F}^{n})^{r}\rightarrow \mathbb {F}$ of degree at most d such that the edges of $\mathcal {H}$ are determined by the zero-patterns of $f_1,\dots ,f_m$ . The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity $(n,d,m)$ has good Ramsey properties, then at least one of the parameters $n,d,m$ must be large. In 2001, Rónyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if G is an algebraic graph of complexity $(n,d,m)$ on N vertices, then either G or its complement contains a complete balanced bipartite graph of size $\Omega _{n,d,m}(N^{1/(n+1)})$ . We extend this result by showing that such G contains either a clique or an independent set of size $N^{\Omega (1/ndm)}$ and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for r-uniform algebraic hypergraphs that are defined by a single polynomial that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.

A novel application on mutually orthogonal graph squares and graph-orthogonal arrays

&lt;abstract&gt; &lt;p&gt;Security of personal information has become a major concern due to the increasing use of the Internet by individuals in the digital world. The main purpose here is to prevent an unauthorized person from gaining access to confidential information. The solution to such a problem is by authentication of users. Authentication has a very important role in achieving security. Mutually orthogonal graph squares (MOGS) are considered the generalization of mutually orthogonal Latin squares (MOLS). Also, MOGS are generated from edge decompositions of complete bipartite graphs by isomorphic graphs. Graph-orthogonal arrays can be constructed by MOGS. In this paper, graph-orthogonal arrays are used for constructing authentication codes. These arrays are the encoding matrices of authentication tags. We introduce the concepts and basic theorems of MOGS, graph-orthogonal arrays, and authentication codes. After constructing graph-orthogonal arrays by MOGS, then there is an established mapping between graph-orthogonal arrays and message set. This manages us to construct perfect non-splitting and splitting Cartesian authentication codes. In both cases, we calculate the probabilities of successful impersonation attacks and substitution attacks. Besides that, the performance of constructed non-splitting and splitting authentication codes is analyzed. In the end, optimal authentication codes and secure authentication codes are constructed.&lt;/p&gt; &lt;/abstract&gt;

Granular computing on basic digraphs

In the present paper we investigate (p, q)-directed complete bipartite graphs ?K p,q, n-directed paths ?Pn and n-directed cycles ?C n from the perspective of Granular Computing. For each model, we establish the general form of all possible indiscernibility relations, analyze the classical rough approximation functions of rough set theory and provide a close formula for the global accuracy average. Finally, we completely determine the attribute dependency function and the global dependency average for both ?C n and ?Kp,q.

Bipartite graphs and best proximity pairs

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.

Anti-Ramsey number of matchings in r-partite r-uniform hypergraphs

An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs H and G, the anti-Ramsey number ar(G,H) of H in G is the maximum number of colors in a coloring of the edges of G so that there does not exist a rainbow copy of H. Li et al. determined the anti-Ramsey number of k-matchings in complete bipartite graphs. Jin and Zang showed the uniqueness of the extremal coloring. In this paper, as a generalization of these results, we determine the anti-Ramsey number arr(Kn1,…,nr,Mk) of k-matchings in complete r-partite r-uniform hypergraphs and show the uniqueness of the extremal coloring. Also, we show that Kk−1,n2,…,nr is the unique extremal hypergraph for Turán number exr(Kn1,…,nr,Mk) and show that arr(Kn1,…,nr, Mk)=exr(Kn1,…,nr,Mk−1)+1, which gives a multi-partite version result of Özkahya and Young's conjecture.

Inertia Sets of Semicliqued Graphs

In this paper, we investigate inertia sets of simple connected undirected graphs. The main focus is on the shape of their corresponding inertia tables, in particular whether or not they are trapezoidal. This paper introduces a special family of graphs created from any given graph, $G$, coined semicliqued graphs and denoted $\widetilde{K}G$. We establish the minimum rank and inertia sets of some $\widetilde{K}G$ in relation to the original graph $G$. For special classes of graphs, $G$, it can be shown that the inertia set of $G$ is a subset of the inertia set of $\widetilde{K}G$. We provide the inertia sets for semicliqued cycles, paths, stars, complete graphs, and for a class of trees. In addition, we establish an inertia set bound for semicliqued complete bipartite graphs.

Higher matching complexes of complete graphs and complete bipartite graphs

For r≥1, the r-matching complex of a graph G, denoted Mr(G), is a simplicial complex whose faces are the subsets H⊆E(G) of the edge set of G such that the degree of any vertex in the induced subgraph G[H] is at most r. In this article, we give a closed form formula for the homotopy type of the (n−2)-matching complex of complete graph on n vertices. We also prove that the (n−1)-matching complex of complete bipartite graph Kn,n is homotopy equivalent to a sphere of dimension (n−1)2−1.

Mutual visibility in graphs

Let G=(V,E) be a graph and P⊆V a set of points. Two points are mutually visible if there is a shortest path between them without further points. P is a mutual-visibility set if its points are pairwise mutually visible. The mutual-visibility number of G is the size of any largest mutual-visibility set. In this paper we start the study about this new invariant and the mutual-visibility sets in undirected graphs. We introduce the Mutual-Visibility problem which asks to find a mutual-visibility set with a size larger than a given number. We show that this problem is NP-complete, whereas, to check whether a given set of points is a mutual-visibility set is solvable in polynomial time. Then we study mutual-visibility sets and mutual-visibility numbers on special classes of graphs, such as block graphs, trees, grids, tori, complete bipartite graphs, cographs. We also provide some relations of the mutual-visibility number of a graph with other invariants.

U-parking functions and (p,q)-parking functions

The notions of U-parking functions and (p,q)-parking functions are two high-dimensional generalizations of the classical parking functions. U-parking functions are defined via a special family of interpolation polynomials called Gončarov polynomials, while (p,q)-parking functions can be interpreted as recurrent configurations in the sandpile model for a complete bipartite graph with an additional root, as introduced by Cori and Poulalhon. In this paper we show that (p,q)-parking functions can be obtained as a specialization of U-parking functions and characterized by a pair of weakly disjoint lattice paths in the grid p×q. Then we present various enumerative results for increasing (p,q)-parking functions.

A Bijection for the Boolean Numbers of Ferrers Graphs

In this note we prove a new characterization of the derangement sets of Ferrers graphs and present a bijection between the derangement sets and F_{lambda }-Callan sequences. In particular, this connection reveals that the boolean numbers of the complete bipartite graphs are the D-relatives of poly-Bernoulli numbers.