Complete Bipartite Graph Research Articles

Row‐Hamiltonian Latin squares and Falconer varieties

AbstractA Latin square is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square is row‐Hamiltonian if the permutation induced by each pair of distinct rows of is a full cycle permutation. Row‐Hamiltonian Latin squares are equivalent to perfect 1‐factorisations of complete bipartite graphs. For the first time, we exhibit a family of Latin squares that are row‐Hamiltonian and also achieve precisely one of the related properties of being column‐Hamiltonian or symbol‐Hamiltonian. This family allows us to construct non‐trivial, anti‐associative, isotopically ‐closed loop varieties, solving an open problem posed by Falconer in 1970.

Constant Sum Group Flows of Graphs

As an analogous concept of a nowhere-zero flow for directed graphs, zero-sum flows and constant-sum flows are defined and studied in the literature. For an undirected graph, a zero-sum flow (constant-sum flow resp.) is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero (constant h resp.), and we call it a zero-sum k-flow (h-sum k-flow resp.) if the values of the edges are less than k. We extend these concepts to general constant-sum A-flow, where A is an Abelian group, and consider the case A=Zk the additive Abelian cyclic group of integer congruences modulo k with identity 0. In the literature, a graph is alternatively called Zk-magic if it admits a constant-sum Zk-flow, where the constant sum is called a magic sum or an index for short. We define the set of all possible magic sums such that G admits a constant-sum Zk-flow to be Ik(G) and call it the magic sum spectrum, or for short, the index set of G with respect to Zk. In this article, we study the general properties of the magic sum spectrum of graphs. We determine the magic sum spectrum of complete bipartite graphs Km,n for m≥n≥2 as the additive cyclic subgroups of Zk generated by kd, where d=gcd(m−n,k). Also, we show that every regular graph G with a perfect matching has a full magic sum spectrum, namely, Ik(G)=Zk for all k≥3. We characterize a 3-regular graph so that it admits a perfect matching if and only if it has a full magic sum spectrum, while an example is given for a 3-regular graph without a perfect matching which has no full magic sum spectrum. Another example is given for a 5-regular graph without a perfect matching, which, however, has a full magic sum spectrum. In particular, we completely determine the magic sum spectra for all regular graphs of even degree. As a byproduct, we verify a conjecture raised by Akbari et al., which claims that every connected 4k-regular graph of even order admits a 1-sum 4-flow. More open problems are included.

On a problem of El-Zahar and Erdős

Two subgraphs A,B of a graph G are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results.We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have large chromatic number: that is, for all t,c≥1 there exists d≥1 such that if G has chromatic number at least d, and does not contain the complete graph Kt as a subgraph, then there are anticomplete subgraphs A,B, where A has minimum degree at least c and B has chromatic number at least c.Second, we look at what happens if we replace the hypothesis that G has sufficiently large chromatic number with the hypothesis that G has sufficiently large minimum degree. This, together with excluding Kt, is not enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding Kt we exclude the complete bipartite graph Kt,t. More exactly: for all t,c≥1 there exists d≥1 such that if G has minimum degree at least d, and does not contain the complete bipartite graph Kt,t as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least c.

On the homotopy type of multipath complexes

AbstractA multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph is the simplicial complex whose faces are the multipaths of . We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.

Method of information technology for structure analysis of urban network fire-rescue units

The subject of this study is the process of analyzing the structure of the network of fire-rescue units of the city in the context of optimizing their spatial distribution. The purpose of this work is to increase the objectivity of decisions made while forming a city network of fire-rescue units by creating an information technology (IT) method for analysis of its structure based on the use of spatially distributed data. Objectives: to find ways to improve the level of fire safety, analyze existing approaches to the formation of a network of fire-rescue units, considering the peculiarities of building and organizing a network, and adapt the classical problem of placement; to propose a method for solving it to minimize the distance between the fire-rescue unit and the possible place of fire while ensuring maximum coverage of the territory by fire service; and to develop IT structure for its implementation based on the information flow model of the process of analyzing the fire-rescue units network using a geospatial approach. The following results were obtained. The study of the classical location problem and its adaptation to real problems arising from the analysis of the urban network of fire-rescue units made it possible to represent it as a set of independent complete bipartite graphs. To search the location of network nodes while solving an adapted problem, an IT method is developed, which, based on the p-median model, combines the author’s methodology for studying information processes and methods of geospatial analysis. Summarizing the requirements of the current legislation, a set of input and output IT data and a set of operations have been formed. The representation of the IT structure in the form of a data flow diagram explains how the set of factors is processed and generalized when making decisions on the creation and / or improvement of the existing city fire department. Conclusions. The results of the bibliographic search confirm the need to consider the spatial features of the area where it is planned to create a fire-rescue unit, as well as the spatial configuration of the urban network of existing fire stations, to evaluate its effectiveness using an integrated indicator. This requires the development of specialized methods focused on the use of geo-information systems for their implementation in decision support systems. Scientific and methodological support for IT has been developed, which gives local authorities a tool for analyzing fire safety in the city to create and / or improve the existing fire protection. An experiment to study the capabilities of the proposed method based on volunteered geographic information on Kharkiv city showed the effectiveness of its use for solving classical problems of placement, considering the accepted restrictions on the spatial availability of fire-rescue units. At the same time, additional opportunities appear in the formation of options for improving the network of fire-rescue units, considering their spatial distribution, workload, accessibility, and the resulting areas of coverage / non-coverage by the fire service. For example, the fire service coverage area of the existing structure of fire stations has been assessed. During the regulated time, it reaches 70% of the Kharkiv city area, and depending on the real road traffic, it can vary from 64.61% to 73.44%. It is illustrated that the creation of two additional fire and rescue units in the northern and southern parts of Kharkiv will increase the coverage area by approximately 5%, on average increasing it to 75.1%

On the study of rainbow antimagic connection number of comb product of tree and complete bipartite graph

Rainbow antimagic coloring is the combination of antimagic labeling and rainbow coloring. The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of [Formula: see text], denoted by [Formula: see text]. Given a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], the function [Formula: see text] from [Formula: see text] to [Formula: see text] is a bijective function. The associated weight of an edge [Formula: see text] under [Formula: see text] is [Formula: see text]. A path [Formula: see text] in the vertex-labeled graph [Formula: see text] is said to be a rainbow [Formula: see text] path if for any two edges [Formula: see text] it satisfies [Formula: see text]. The function [Formula: see text] is called a rainbow antimagic labeling of [Formula: see text] if there exists a rainbow [Formula: see text] path for every two vertices [Formula: see text]. When we assign each edge [Formula: see text] with the color of the edge weight [Formula: see text], we say the graph [Formula: see text] admits a rainbow antimagic coloring. In this paper, we show the new lower bound and exact value of the rainbow antimagic connection number of comb product of any tree and complete bipartite graph.

On minimum spanning trees for random Euclidean bipartite graphs

AbstractWe consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$ -th power of their Euclidean distance, with $p\gt 0$ . In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$ , we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$ , we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

Occupancy probabilities in superintegrable bosonic networks

We associate a bosonic network to each complete bipartite graph. A Hamiltonian is defined with hopping terms on the edges of the graph, and a global interaction term depending on the vertex sets. For a generic graph this Hamiltonian is superintegrable, and we derive a Bethe Ansatz solution for the energy and eigenstates. These results provide the means to investigate the quantum dynamics and allow for the computation of occupancy probabilities in certain regimes. We use these results to gain an understanding of entanglement evolution in the network.

The saturation number of K3,3

A graph G is called F-saturated if G does not contain F as a subgraph (not necessarily induced) but the addition of any missing edge to G creates a copy of F. The saturation number of F, denoted by sat(n,F), is the minimum number of edges in an n-vertex F-saturated graph. Determining the saturation number of complete bipartite graphs is one of the most important problems in the study of saturation numbers. The value of sat(n,K2,2) was shown to be ⌊3n−52⌋ by Ollmann, and a shorter proof was later given by Tuza. For K2,3, there has been a series of study aiming to determine sat(n,K2,3) over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that sat(n,K2,3)=2n−3 for all n≥5. Pikhurko and Schmitt conjectured that sat(n,K3,3)=(3+o(1))n. In this paper, for n≥9, we give an upper bound of 3n−9 for sat(n,K3,3), and prove that 3n−9 is also a lower bound when the minimum degree of a K3,3-saturated graph is 2 or 5, where it is trivial when the minimum degree is greater than 5.

Induced subgraphs and tree decompositions V. one neighbor in a hole

AbstractWhat are the unavoidable induced subgraphs of graphs with large treewidth? It is well‐known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the “basic treewidth obstructions”). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so‐called “layered wheels,” contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

Finding Large Rainbow Trees in Colourings of $K_{n,n}$

A subgraph of an edge-coloured graph is called rainbow if all of its edges have distinct colours. An edge-colouring is called locally $k$-bounded if each vertex is incident with at most $k$ edges of the same colour. Recently, Montgomery, Pokrovskiy and Sudakov showed that for large $n$, a certain locally 2-bounded edge-colouring of the complete graph $K_{2n+1}$ contains a rainbow copy of any tree with $n$ edges, thereby resolving a long-standing conjecture by Ringel: For large $n$, $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges. In this paper, we employ their methods to show that any locally $k$-bounded edge-colouring of the complete bipartite graph $K_{n,n}$ contains a rainbow copy of any tree $T$ with $(1- o(1))n/k$ edges. We show that this implies that every tree with $n$ edges packs at least $n$ times into $K_{n+o(1),n+o(1)}$. We conjecture that for large $n$, $K_{n,n}$ can be decomposed into $n$ copies of any tree with $n$ edges.

Intelligent team formation and player selection: a data-driven approach for football coaches

This article presents a data-driven approach called Intelligent Team Formation and Player Selection (ITFPS), aimed at assisting football coaches in making informed decisions about team formation and player selection. The proposed approach utilizes deep neural networks to evaluate the suitability of each player for different positions on the field. The problem is then formulated as finding the maximum weighted matching in a complete bipartite graph, with the objective of achieving the best possible alignment between team members and the positions designated by the coach. The Hungarian algorithm is employed to facilitate this matching process. Furthermore, the approach allows coaches to select a system of distinct representatives for each position, based on the specific qualities expected from players in a given match. The effectiveness of the approach is demonstrated through tests conducted on four teams from the 2021–2022 English Premier League. The results indicate that ITFPS produces decisions comparable to those made by successful coaches. By optimizing team formations and enabling the utilization of rotating formations, this approach not only enhances team performance but also empowers coaches to make strategic decisions while fully leveraging the potential of their players. ITFPS serves as an intelligent assistant for coaches, complementing their strategic perspectives.

Induced subgraphs and tree decompositions VII. Basic obstructions in H-free graphs

We say a class C of graphs is clean if for every positive integer t there exists a positive integer w(t) such that every graph in C with treewidth more than w(t) contains an induced subgraph isomorphic to one of the following: the complete graph Kt, the complete bipartite graph Kt,t, a subdivision of the (t×t)-wall or the line graph of a subdivision of the (t×t)-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all H-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph H) is clean if and only if H is a forest whose components are subdivided stars.Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest H as above, we show that forbidding certain connected graphs containing H as an induced subgraph (rather than H itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer η, a complete description of unavoidable connected induced subgraphs of a connected graph G containing η vertices from a suitably large given set of vertices in G. This is of independent interest, and will be used in subsequent papers in this series.

Spectral approach to quantum searching on Markov chains—the complete bipartite graph

Spectral approach to quantum searching on Markov chains—the complete bipartite graph

On the choosability of -minor-free graphs

AbstractGiven a graph $H$ , let us denote by $f_\chi (H)$ and $f_\ell (H)$ , respectively, the maximum chromatic number and the maximum list chromatic number of $H$ -minor-free graphs. Hadwiger’s famous colouring conjecture from 1943 states that $f_\chi (K_t)=t-1$ for every $t \ge 2$ . A closely related problem that has received significant attention in the past concerns $f_\ell (K_t)$ , for which it is known that $2t-o(t) \le f_\ell (K_t) \le O(t (\!\log \log t)^6)$ . Thus, $f_\ell (K_t)$ is bounded away from the conjectured value $t-1$ for $f_\chi (K_t)$ by at least a constant factor. The so-called $H$ -Hadwiger’s conjecture, proposed by Seymour, asks to prove that $f_\chi (H)={\textrm{v}}(H)-1$ for a given graph $H$ (which would be implied by Hadwiger’s conjecture).In this paper, we prove several new lower bounds on $f_\ell (H)$ , thus exploring the limits of a list colouring extension of $H$ -Hadwiger’s conjecture. Our main results are:For every $\varepsilon \gt 0$ and all sufficiently large graphs $H$ we have $f_\ell (H)\ge (1-\varepsilon )({\textrm{v}}(H)+\kappa (H))$ , where $\kappa (H)$ denotes the vertex-connectivity of $H$ .For every $\varepsilon \gt 0$ there exists $C=C(\varepsilon )\gt 0$ such that asymptotically almost every $n$ -vertex graph $H$ with $\left \lceil C n\log n\right \rceil$ edges satisfies $f_\ell (H)\ge (2-\varepsilon )n$ .The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of $H$ -minor-free graphs is separated from the desired value of $({\textrm{v}}(H)-1)$ by a constant factor for all large graphs $H$ of linear connectivity. The second result tells us that for almost all graphs $H$ with superlogarithmic average degree $f_\ell (H)$ is separated from $({\textrm{v}}(H)-1)$ by a constant factor arbitrarily close to $2$ . Conceptually these results indicate that the graphs $H$ for which $f_\ell (H)$ is close to the conjectured value $({\textrm{v}}(H)-1)$ for $f_\chi (H)$ are typically rather sparse.

Positional difference and Frequency (PdF) based alignment-free technique for genome sequence comparison

In the field of computational biology, genome sequence comparison among different species is essential and has applications in both the research and scientific fields. Owing to the lengthy processing time and large number of data sets, the alignment-based approaches are unsuitable and ineffective. Therefore, alignment-free techniques have obtained popularity for acquiring proper sequence clustering and evolutionary relationship among species. In this paper, a complete bipartite graph based Positional difference and Frequency (PdF) vector descriptor is introduced. Positional difference and Frequency, two parameters, are applied to the genome sequence to create a 16- dimensional vector descriptor using the di-nucleotide representation of genome sequence. Subsequently, a distance matrix is calculated to construct the phylogenetic trees for different data sets of mammals and viruses. The achieved outcomes are compared with the phylogenetic trees of the earlier methods viz. the FFP method, the ClustalW method, the MEV method, the PCNV method and the FIS method. In most instances, the proposed method produces more precise outcomes than the preceding techniques and has potential for genome sequence comparison on both the equal and unequal length of data-sets. Communicated by Ramaswamy H. Sarma

INFORMATION TECHNOLOGY OF FORMING THE EDUCATIONAL NETWORK OF THE TERRITORIAL COMMUNITY

Context. Local government organizations have significant decision-making power in the field of education. This requires the development of understandable tools that help form a network of educational institutions that provide high-quality educational services. The object of the study was the process of forming an educational network (ENW) of the community in the implementation of territorial development projects as part of the reform of local self-government.
 Objective. The goal of the work is to increase objectivity of decisions made in the formation of the ENW territorial community, summarizing demographic, infrastructural, personnel, economic factors when choosing the location of ENW facilities.
 Method. The study of the classical placement problem and its further adaptation to real problems arising from the implementation of the education reform made it possible to represent ENW territorial communities as a set of independent complete bipartite graphs. In this case, to solve the problem of choosing the location of an educational institution on the network, an information technology (IT) was developed to form an ENW territorial community. Based on the adapted p-median model and methods of geospatial analysis, generalizing the requirements of the current legislation, a set of input and output parameters of IT and a set of its operations are formed. The representation of the IT structure in the form of an IDEF0-model clearly explains how a combination of various factors is processed and generalized when making decisions while creating ENW and looking for ways to improve it.
 Results. The developed IT was investigated in solving the problem of forming a network of lyceums in the Kharkiv region using geospatial information, open statistical data and data on lyceums. The proposed accommodation options make it possible to achieve a general level of accessibility of specialized secondary education of almost 94%. IT has additional tools for solving the problems of transporting students to the place of study and home.
 Conclusions. The experiments carried out confirmed the operability of the proposed IT. The generalization of the results obtained makes it possible to recommend it for practical use in solving the problems of analyzing the current state of ENW, finding ways to improve it and possible directions for development, as well as evaluating solutions related to the spatial planning of ENW.

Counterexamples to Gerbner's conjecture on stability of maximal F‐free graphs

AbstractLet and let be an ‐chromatic graph with a color‐critical edge, that is, there exists such that has chromatic number . Gerbner recently conjectured that every ‐vertex maximal ‐free graph with at least edges contains an induced complete ‐partite graph on vertices. Let be a graph obtained from copies of by sharing a common edge. In this paper, we show that for all if is an ‐vertex maximal ‐free graph with at least edges, then contains an induced complete bipartite graph on vertices. We also construct graphs to show that the magnitude in is best possible, which disproves Gerbner's conjecture for .

10 problems for partitions of triangle-free graphs

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erdős’ Sparse Half Conjecture.Among others we prove the following variant of it: For every sufficiently large even integer n the following holds. Every triangle-free graph on n vertices has a partition V(G)=A∪B with |A|=|B|=n/2 such that e(G[A])+e(G[B])≤n2/16. This result is sharp since the complete bipartite graph with class sizes 3n/4 and n/4 achieves equality, when n is a multiple of 4.Additionally, we discuss similar problems for K4-free graphs.

Grundy number of corona product of some graphs

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex u colored with color i, 1 ≤ i ≤ k, is adjacent to i – 1 vertices colored with each color j, 1 ≤ j ≤ i − 1. In this paper we obtain the Grundy number of corona product of some graphs, denoted by G ◦ H. First, we consider the graph G be 2-regular graph and H be a cycle, complete bipartite, ladder graph and fan graph. Also we consider the graph G and H be a complete bipartite graphs, fan graphs.