Hexagonal Graphs Research Articles

Schultz and Modified Schultz Polynomials of Edges Induce Chain and Ring for Hexagonal Graphs

Schultz polynomial is one of the must significant formulas that represent a relationship between the degree’s of vertices in a simple connected graph G and the distances between these vertices. In this work, Schultz and modified Schultz polynomials, as well as their topological indices of chain and ring hexagonal graphs, have been successfully identified.

Optimized Evaluation of Mobile Base Station by Modern Topological Invariants

Due to a tremendous increase in mobile traffic, mobile operators have started to restructure their networks to offload their traffic. New research directions will lead to fundamental changes in the design of future Fifth-generation (5G) cellular networks. For the formal reason, the study solves the physical network of the mobile base station for the prediction of the best characteristics to develop an enhanced network with the help of graph theory. Any number that can be uniquely calculated by a graph is known as a graph invariant. During the last two decades, innumerable numerical graph invariants have been portrayed and used for correlation analysis. In any case, no efficient assessment has been embraced to choose, how much these invariants are connected with a network graph. This paper will talk about two unique variations of the hexagonal graph with great capability of forecasting in the field of optimized mobile base station topology in setting with physical networks. Since K-banhatti sombor invariants (KBSO) and Contrharmonic-quadratic invariants (CQIs) are newly introduced and have various expectation characteristics for various variations of hexagonal graphs or networks. As the hexagonal networks are used in mobile base stations in layered, forms called honeycomb. The review settled the topology of a hexagon of two distinct sorts with two invariants KBSO and CQIs and their reduced forms. The deduced outcomes can be utilized for the modeling of mobile cellular networks, multiprocessors interconnections, microchips, chemical compound synthesis and memory interconnection networks. The results find sharp upper bounds and lower bounds of the honeycomb network to utilize the Mobile base station network (MBSN) for the high load of traffic and minimal traffic also.

3-path vertex cover and dissociation number of hexagonal graphs

A subset P of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from P. The cardinality of a minimum k-path vertex cover is called the k-path vertex cover number of G, and is denoted by ?k(G). It is known that the problem of finding a minimum 3-path vertex cover is NP-hard for planar graphs. In this paper we establish an upper bound for ?3(G), where G is from an important family of planar graphs, called hexagonal graphs, arising from real world applications.

Super Fibonacci Graceful Graphs and Fibonacci Cubes

The popularity of Fibonacci cubes is due to their wide range of uses. In mathematical chemistry, this concept is used in the study of hexagonal graphs. In computer science, Fibonacci cubes are interesting from an algorithmic point of view. V. Hsu introduced them in 1993 to simulate the connections of multiprocessor computer networks. He wanted to get graphs with hypercube properties, the order of which is not a power of two. Therefore, the problem of embedding other graphs in Fibonacci cubes is of interest.

Narrow peaks of full transmission in simple quantum graphs

This work deals with quantum graphs, focusing on the transmission properties they engender. We first select two simple diamond graphs, and two hexagonal graphs in which the vertices are all of degree 3, and investigate their transmission coefficients. In particular, we identified regions in which the transmission is fully suppressed. We also considered the transmission coefficients of some series and parallel arrangements of the two basic graphs, with the vertices still preserving the degree 3 condition, and then identified specific series and parallel compositions that allow for windows of no transmission. Inside some of these windows, we found very narrow peaks of full transmission, which are consequences of constructive quantum interference. Possibilities of practical use as the experimental construction of devices of current interest to control and manipulate quantum transmission are also discussed.

Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs

On the space of Ising configurations on the 2-d square lattice, we consider a family of non Gibbsian measures introduced by using a pair Hamiltonian, depending on an additional inertial parameter $q$. These measures are related to the usual Gibbs measure on $\Z^2$ and turn out to be the marginal of the Gibbs measure of a suitable Ising model on the hexagonal lattice. The inertial parameter $q$ tunes the geometry of the system. The critical behaviour and the decay of correlation functions of these measures are studied thanks to relation with the Random Cluster model.

Cycle super magic labeling of pumpkin, octagonal and hexagonal graphs

A simple graph G(V, E) admits a H-covering, if every edge in E(G) belongs to a subgraph of G isomorphic to H. The graph G is said to be H-magic, if there exists a bijection ψ: V(G) ∪ E(G) → {1, 2, 3, … ,|V(G)|+|E(G)|} such that for every subgraph H¢ of G isomorphic to is constant. Moreover G is said to be H-super magic, if ψ (V(G)) = {1, 2, 3, … ,|V(G)|}. In this paper, we study the cycle-super magic labeling of a pumpkin graph and two classes of planar maps containing 8-sided and 4-sided faces or 6-sided and 4-sided faces, respectively.

The crossing number of hexagonal graph H3,n in the projective plane

The crossing number of hexagonal graph <i>H_3,n</i> in the projective plane

The crossing number of the hexagonal graph H_{3,n}

The crossing number of the hexagonal graph H_{3,n}

Maximum Induced Matching of Hexagonal Graphs

A matching in a graph is a set of edges no two of which share a common vertex. A matching is an induced matching if no two edges in the matching have a third edge in the graph connecting them. The problem of finding a maximum induced matching or shortly MIM is known to be NP-hard in general, and it remains so even when the input graph is bipartite. The decision problem of MIM is NP-complete in general, and it remains NP-complete even if restricted to several classes of graphs. On the other hand, the problem has been shown to be polynomial for some special sets of graphs. In this paper, we give tight upper and lower bounds on maximum induced matching in special subset of planar graphs, called hexagonal graphs.

Online multi-coloring with advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. For hexagonal graphs, negative results trivially carry over to 3-colorable graphs, and most of our positive results do as well. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.

Multicoloring of cannonball graphs

The frequency allocation problem that appeared in the design of cellular telephone networks can be regarded as a multicoloring problem on a weighted hexagonal graph, which opened some still interesting mathematical problems. We generalize the multicoloring problem into higher dimension and present the first approximation algorithms for multicoloring of the so called cannonball graphs.

평면의 채색수 알고리즘

본 논문은 평면상의 거리가 1인 인접 정점들에 대해 서로 다른 색을 칠할 경우 최대로 필요한 색인 채색수를 찾는 문제를 연구하였다. 지금까지 채색수 상한 값은 <TEX>$4{\leq}{\chi}(G){\leq}7$</TEX>로 알려져 있으며, Hadwiger-Nelson은 <TEX>${\chi}(G){\leq}7$</TEX>, Soifer는 <TEX>${\chi}(G){\leq}9$</TEX>를 제안하였다. 먼저, 최소로 필요로 하는 채색수를 구하는 알고리즘을 제안하고, Hadwiger-Nelson의 정육각형 그래프를 대상으로 채색수를 구한 결과 <TEX>${\chi}(G)=3$</TEX>이 될 수 있음을 보였다. Hadwiger-Nelson의 정육각형 그래프를 12개 인접 정점으로 가정할 경우 <TEX>${\chi}(G)=4$</TEX>를 구하였다. 또한, Soifer의 8개 인접 정점 정사각형 그래프에 대해 채색수를 구한 결과 <TEX>${\chi}(G)=4$</TEX>임을 보였다. 결국, 제안된 알고리즘은 최소 차수 정점부터 색을 배정하는 단순한 다항시간 규칙을 적용하여 평면의 최대 채색수는 <TEX>${\chi}(G)=4$</TEX>임을 제안한다. In this paper, I seek the chromatic number, the maximum number of colors necessary when adjoining vertices in the plane separated apart at the distance of 1 shall receive distinct colors. The upper limit of the chromatic number has been widely accepted as <TEX>$4{\leq}{\chi}(G){\leq}7$</TEX> to which Hadwiger-Nelson proposed <TEX>${\chi}(G){\leq}7$</TEX> and Soifer <TEX>${\chi}(G){\leq}9$</TEX> I firstly propose an algorithm that obtains the minimum necessary chromatic number and show that <TEX>${\chi}(G)=3$</TEX> is attainable by determining the chromatic number for Hadwiger-Nelson's hexagonal graph. The proposed algorithm obtains a chromatic number of <TEX>${\chi}(G)=4$</TEX> assuming a Hadwiger-Nelson's hexagonal graph of 12 adjoining vertices, and again <TEX>${\chi}(G)=4$</TEX> for Soifer's square graph of 8 adjoining vertices. assert. Based on the results as such that this algorithm suggests the maximum chromatic number of a planar graph is <TEX>${\chi}(G)=4$</TEX> using simple assigned rule of polynomial time complexity to color for a vertex with minimum degree.

1-local 33/24-competitive Algorithm for Multicoloring Hexagonal Graphs

Graph Theory In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by assigned frequencies, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where each vertex knows its position in the graph. We present a 1-local 33/24-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 7/5-competitive algorithm.

A linear time algorithm for 7-[3]coloring triangle-free hexagonal graphs

Given a graph G and p∈N, a proper n-[p]coloring is a mapping f:V(G)→2{1,…,n} such that |f(v)|=p for any vertex v∈V(G) and f(v)∩f(u)=∅ for any pair of adjacent vertices u and v. An n-[p]coloring is a special case of a multicoloring. Finding a multicoloring of induced subgraphs of the triangular lattice (called hexagonal graphs) has important applications in cellular networks. In this article we provide an algorithm to find a 7-[3]coloring of triangle-free hexagonal graphs in linear time, without using the four-color theorem, which solves the open problem stated by Sau, Šparl and Žerovnik (2011) and improves the result of Sudeep and Vishwanathan (2005), who proved the existence of a 14-[6]coloring.

A 1-local 4/3-competitive algorithm for multicoloring a subclass of hexagonal graphs

We consider a frequency allocation problem, in which we are given a cellular telephone network whose geographical coverage area is divided into cells in which phone calls are serviced by frequencies assigned to them, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e., to minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph. In this paper, we present a 1-local 4/3-competitive distributed algorithm for multicoloring a hexagonal graph without certain forbidden configuration (introduced in Šparl and Žerovnik (2010) [7]).

1-Local 7/5-Competitive Algorithm for Multicoloring Hexagonal Graphs

In the frequency allocation problem, we are given a cellular telephone network whose geographical coverage area is divided into cells, where phone calls are serviced by frequencies assigned to them, so that none of the pairs of calls emanating from the same or neighboring cells is assigned the same frequency. The problem is to use the frequencies efficiently, i.e. minimize the span of frequencies used. The frequency allocation problem can be regarded as a multicoloring problem on a weighted hexagonal graph, where every vertex knows its position in the graph. We present a 1-local 7/5-competitive distributed algorithm for multicoloring a hexagonal graph, thereby improving the previous 1-local 17/12-competitive algorithm.

Simpler multicoloring of triangle-free hexagonal graphs

Given a graph G and a demand function p:V(G)→N, a proper n-[p]coloring is a mapping f:V(G)→2{1,…,n} such that |f(v)|≥p(v) for every vertex v∈V(G) and f(v)∩f(u)=0̸ for any two adjacent vertices u and v. The least integer n for which a proper n-[p]coloring exists, χp(G), is called multichromatic number of G. Finding multichromatic number of induced subgraphs of the triangular lattice (called hexagonal graphs) has applications in cellular networks. Weighted clique number of a graph G, ωp(G), is the maximum weight of a clique in G, where the weight of a clique is the total demand of its vertices. McDiarmid and Reed (2000) [8] conjectured that χp(G)≤(9/8)ωp(G)+C for triangle-free hexagonal graphs, where C is some absolute constant. In this article, we provide an algorithm to find a 7-[3]coloring of triangle-free hexagonal graphs (that is, when p(v)=3 for all v∈V(G)), which implies that χp(G)≤(7/6)ωp(G)+C. Our result constitutes a shorter alternative to the inductive proof of Havet (2001) [5] and improves the short proof of Sudeep and Vishwanathan (2005) [17], who proved the existence of a 14-[6]coloring. (It has to be noted, however, that our proof makes use of the 4-color theorem.) All steps of our algorithm take time linear in |V(G)|, except for the 4-coloring of an auxiliary planar graph. The new techniques may shed some light on the conjecture of McDiarmid and Reed (2000) [8].

One-regular graphs of square-free order of prime valency

A graph is one-regular if its automorphism group acts regularly on the set of arcs of the graph. Marušič and Pisanski [D. Marušič and T. Pisanski, Symmetries of hexagonal graphs on the torus, Croat. Chemica Acta 73 (2000) 969–981] classified one-regular Cayley graphs on a dihedral group of valency 3, and Kwak et al. [J.H. Kwak, Y.S. Kwon, J.M. Oh, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, J. Combin. Theory B 98 (2008) 585–598] classified those of valency 5. In this paper one-regular Cayley graphs on a dihedral group of any prime valency are classified and enumerated. It is shown that for an odd prime q , there exists a q -valent one-regular Cayley graph on the dihedral group of order 2 m if and only if m = q t p 1 e 1 p 2 e 2 ⋯ p s e s ≥ 13 , where t ≤ 1 , s ≥ 1 , e i ≥ 1 and p i ’s are distinct primes such that q ∣ ( p i − 1 ) . There are exactly ( q − 1 ) s − 1 non-isomorphic such graphs for a given order. Consequently, one-regular cyclic Haar graphs of prime valency are classified and enumerated. Furthermore, it is shown that every q -valent one-regular graph of square-free order is a Cayley graph on a dihedral group, and as a result, q -valent one-regular graphs of square-free order are classified and enumerated.

IMPROPER COLORING OF WEIGHTED GRID AND HEXAGONAL GRAPHS

We study a weighted improper coloring problem motivated by a frequency allocation problem. It consists of associating to each vertex a set of p(v) (weight) distinct colors (frequencies), such that the set of vertices having a given color induces a graph of degree at most k (the case k = 0 corresponds to proper coloring). The objective is to minimize the number of colors. We propose approximation algorithms to compute such a coloring for general graphs. We apply these to obtain good approximation ratio for grid and hexagonal graphs. Furthermore we give exact results for the 2-dimensional grid and the triangular lattice when the weights are all the same.