Complete Bipartite Graph Kn Research Articles

Kirchhoff index in line, subdivision and total graphs of a regular graph

Let G be a connected regular graph and l(G), s(G), t(G) the line, subdivision, total graphs of G, respectively. In this paper, we derive formulae and lower bounds of the Kirchhoff index of l(G), s(G) and t(G), respectively. In particular, we give special formulae for the Kirchhoff index of l(G), s(G) and t(G), where G is a complete graph Kn, a regular complete bipartite graph Kn,n and a cycle Cn.

Symmetric Hamilton cycle decompositions of complete graphs minus a 1-factor

Let n≥2 be an integer. The complete graph Kn with a 1-factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that Kn−F has a decomposition into Hamilton cycles which are symmetric with respect to the 1-factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph Kn, n has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1-factor of Kn, n, then Kn, n−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1-15, 2010

GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

AbstractWe consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponentn– their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graphKn,n, wherenis an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphsKn,n[7] and generalises the analogous results for maps obtained in [9], partly using different methods.

Mutually orthogonal graph squares

AbstractA decomposition 𝒢={G1, G2,…,Gs} of a graph G is a partition of the edge set of G into edge‐disjoint subgraphs G1, G2,…,Gs. If Gi≅H for all i∈{1, 2, …, s}, then 𝒢 is a decomposition of G by H. Two decompositions 𝒢={G1, G2, …, Gn} and ℱ={F1, F2,…,Fn} of the complete bipartite graph Kn,n are orthogonal if |E(Gi)∩E(Fj)|=1 for all i,j∈{1, 2, …, n}. A set of decompositions {𝒢1, 𝒢2, …, 𝒢k} of Kn, n is a set of k mutually orthogonal graph squares (MOGS) if 𝒢i and 𝒢j are orthogonal for all i, j∈{1, 2, …, k} and i≠j. For any bipartite graph G with n edges, N(n, G) denotes the maximum number k in a largest possible set {𝒢1, 𝒢2, …, 𝒢k} of MOGS of Kn, n by G. El‐Shanawany conjectured that if p is a prime number, then N(p, Pp+ 1)=p, where Pp+ 1 is the path on p+ 1 vertices. In this article, we prove this conjecture. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 369–373, 2009

Regular embeddings ofKn,nwherenis a power of 2. I: Metacyclic case

A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs Kn,n, where n=2e. The method involves groups G which factorize as a product XY of two cyclic groups of order n so that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G. Employing the classification we investigate automorphisms of these groups, resulting in a classification of regular embeddings of Kn,n based on that for G. We prove that given n=2e (for e≥3), there are, up to map isomorphism, exactly 2e−2+4 regular embeddings of Kn,n. Our analysis splits naturally into two cases depending on whether the group G is metacyclic or not.

Regular embeddings ofKn,nwherenis an odd prime power

We show that if n=pe where p is an odd prime and e≥1, then the complete bipartite graph Kn,n has pe−1 regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type {2n,n} and genus (n−1)(n−2)/2; one is reflexible, and the rest are chiral. The method involves groups which factorise as a product of two cyclic groups of order n. We deduce that if n is odd then Kn,n has at least n/∏p|np orientable regular embeddings, and that this lower bound is attained if and only if no two primes p and q dividing n satisfy p≡1mod(q).

Regular orientable embeddings of complete bipartite graphs

AbstractIn this paper, it will be shown that the isomorphism classes of regular orientable embeddings of the complete bipartite graph Kn,n are in one‐to‐one correspondence with the permutations on n elements satisfying a given criterion, and the isomorphism classes of them are completely classified when n is a product of any two (not necessarily distinct) prime numbers. For other n, a lower bound of the number of those isomorphism classes of Kn,n is obtained. As a result, many new regular orientable embeddings of the complete bipartite graph are constructed giving an answer of Nedela‐Škoviera's question raised in 12. © 2005 Wiley Periodicals, Inc. J Graph Theory

On the Number of Bases of Bicircular Matroids

Let t(G) be the number of spanning trees of a connected graph G, and let b(G) be the number of bases of the bicircular matroid B(G). In this paper we obtain bounds relating b(G) and t(G), and study in detail the case where G is a complete graph Kn or a complete bipartite graph Kn,m.

Optimal accumulation of Jacobian matrices by elimination methods on the dual computational graph

The accumulation of the Jacobian matrix F’ of a vector function *** equation here *** can be regarded as a transformation of its linearized computational graph into a subgraph of the directed complete bipartite graph Kn,m. This transformation can be performed by applying different elimination techniques that may lead to varying costs for computing F’. This paper introduces face elimination as the basic technique for accumulating Jacobian matrices by using a minimal number of arithmetic operations. Its superiority over both edge and vertex elimination methods is shown. The intention is to establish the conceptual basis for the ongoing development of algorithms for optimizing the computation of Jacobian matrices.

Ramsey-Type Theorems for Spatial Graphs and Good Drawings

We shall prove that for any spatial graphH, there exists a pair of natural numbers (N, M) such that any spatial embedding of the complete bipartite graphKN, Mwhose projection is a good drawing on the plane contains a subgraph which is ambient isotopic to a subdivision ofH.

Structural Properties of Context-Free Sets of Graphs Generated by Vertex Replacement

We establish that a Vertex Replacement set of graphs, i.e., a set of graphs generated by a C-edNCE or, equivalently, by a separated handle rewriting graph grammar is Hyperedge Replacement, i.e., is generated by a hyperedge replacement graph grammar, iff its graphs do not contain arbitrary large complete bipartite graphs Kn, n as subgraphs. Another equivalent condition is that its graphs have a number of edges that is linearly bounded in terms of the number of vertices. These properties are decidable by means of an appropriate extension of the theorem by Parikh that characterizes the commutative images of context-free languages. We extend these results to hypergraphs.

On random minimum length spanning trees

We extend and strengthen the result that, in the complete graphKn with independent random edge-lengths uniformly distributed on [0, 1], the expected length of the minimum spanning tree tends toζ(3) asn→∞. In particular, ifKn is replaced by the complete bipartite graphKn, n then there is a corresponding limit of 2ζ (3).

On the contact dimensions of graphs

Every simple graphG=(V, E) can be represented by a family of equal nonoverlapping spheres {Sv:v � V} in a Euclidean spaceRn in such a way thatuv � E if and only ifSu andSv touch each other. The smallest dimensionn needed to representG in such a way is called the contact dimension ofG and it is denoted by cd(G). We prove that (1) cd(T)<(7.3) log |T| for every treeT, and (2) $$m - 1 + \frac{n}{2}\left( {1 - \frac{n}{{2\pi m}}\left( {\sqrt {\frac{{n + 4\pi m}}{n}} - 1} \right)} \right)< cd(K_m + E_n ) \leqslant m - 1 + \left\lceil {\frac{n}{2}} \right\rceil ,$$ whereKm+En is the join of the complete graph of orderm and the empty graph of ordern. For the complete bipartite graphKn,n this implies (1.286)n�1

An analogue of the Hadamard conjecture for n × n matrices with n ≡2 (mod 4)

AbstractIt is known that the problem of settling the existence of an n × n Hadamard matrix, where n is divisible by 4, is equivalent to that of finding the cardinality of a smallest set T of 4-circuits in the complete bipartite graph Kn, n such that T contains at least one circuit of each copy of K2,3 in Kn, n. Here we investigate the case where n ≡ 2 (mod 4), and we show that the problem of finding the cardinality of T is equivalent to that of settling the existence of a certain kind of n × n matrix. Moreover, we show that the case where n ≡ 2 (mod 4) differs from that where n ≡ 0 (mod 4) in that the problem of finding the cardinality of T is not equivalent to that of maximising the determinant of an n × n (1,-1)-matrix.