A problem concerning infinite graphs

We show that there are graphs G and H which satisfy: (I) for every integer n, H contains n disjoint graphs each isomorphic to G, and (II) H does not contain infinitely many disjoint graphs each isomorphic to G. This answers one of the questions raised by Halin in the Graph Theory Newsletter.     

On a problem of R. Halin concerning infinite graphs II

For every countable, connected graph A containing no one-way infinite path the following is shown: Let G be an arbitrary graph which contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G also contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. In addition, corrections of errors are given that occur unfortunately in the forerunner of the present paper.     

Über eine eigenschaft lokalfiniter,unendlicher bäume

Let A be an arbitrary locally finite, infinite tree and assume that a graph G contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. This sharpens a theorem of Halin [5], who proved the corresponding result for the case that A is a tree in which each vertex has degree not greater than 3.     

G-decomposition of Kn, where G has four vertices or less

The complete graph K_n, is said to have a G-decomposition if it is the union of edge disjoint subgraphs each isomorphic to G. The set of values of n for which K_n has a G-decomposition is determined if G has four vertices or less.     

A Complete Solution to Spectrum Problem for Five‐Vertex Graphs with Application to Traffic Grooming in Optical Networks

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. Grooming uniform all‐to‐all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph on n vertices into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The existence spectrum problem of G‐designs for five‐vertex graphs is a long standing problem posed by Bermond, Huang, Rosa and Sotteau in 1980, which is closely related to traffic groomings in optical networks. Although considerable progress has been made over the past 30 years, the existence problems for such G‐designs and their related traffic groomings in optical networks are far from complete. In this paper, we first give a complete solution to this spectrum problem for five‐vertex graphs by eliminating all the undetermined possible exceptions. Then, we determine almost completely the minimum drop cost of 8‐groomings for all orders n by reducing the 37 possible exceptions to 8. Finally, we show the minimum possible drop cost of 9‐groomings for all orders n is realizable with 14 exceptions and 12 possible exceptions.     

Subgraphs as circuits and bases of matroids

As is well known, the cycles of any given graph G may be regarded as the circuits of a matroid defined on the edge set of G. The question of whether other families of connected graphs exist such that, given any graph G, the subgraphs of G isomorphic to some member of the family may be regarded as the circuits of a matroid defined on the edge set of G led us, in two other papers, to the proof of some results concerning properties of the cycles when regarded as circuits of such matroids. Here we prove that the wheels share many of these properties with the cycles. Moreover, properties of subgraphs which may be regarded as bases of such matroids are also investigated.     

The new methods for constructing matching-equivalence graphs

Two graphs G and H with order n are said to be matching-equivalent if and only if the number of r-matchings (i.e., the number of ways in which r disjoint edges can be chosen) is the same for each of the graphs G and H for each r, where 0?r?n. In this paper, the new methods for constructing ‘matching-equivalent’ graphs are given, and some families of non-matching unique graphs are also obtained.     

A Note on a Packing Problem in Transitive Tournaments

Let TT_n be a transitive tournament on n vertices. We show that for any directed acyclic graph G of order n and of size not greater than two directed graphs isomorphic to G are arc disjoint subgraphs of TT_n. Moreover, this bound is generally the best possible. The research partially supported by KBN grant 2 P03A 016 18     

A survey on the existence of G‐Designs

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G‐designs. This includes references to all the necessary designs and constructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 373–410, 2008     

Covering graphs by the minimum number of equivalence relations

An equivalence graph is a vertex disjoint union of complete graphs. For a graphG, let eq(G) be the minimum number of equivalence subgraphs ofG needed to cover all edges ofG. Similarly, let cc(G) be the minimum number of complete subgraphs ofG needed to cover all its edges. LetH be a graph onn vertices with maximal degree ≦d (and minimal degree ≧1), and letG= \(\bar H\) be its complement. We show that $$\log _2 n - \log _2 d \leqq eq(G) \leqq cc(G) \leqq 2e^2 (d + 1)^2 \log _e n.$$ The lower bound is proved by multilinear techniques (exterior algebra), and its assertion for the complement of ann-cycle settles a problem of Frankl. The upper bound is proved by probabilistic arguments, and it generalizes results of de Caen, Gregory and Pullman.     

On cubical graphs

It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of n-dimensional cubes. This has found applications, for example, in coding theory, data transmission, and linguistics. In this note, we study the structure of those graphs G, called cubical graphs (not to be confused with cubic graphs, those graphs for which all vertices have degree 3), which can be embedded into an n-dimensional cube. A basic technique used is the investigation of graphs which are critically nonembeddable, i.e., which can not be embedded but all of whose subgraphs can be embedded.     

Self-orthogonal decompositions of graphs into matchings

Given a simple graph H, a self-orthogonal decomposition (SOD) of H is a collection of subgraphs of H, all isomorphic to some graph G, such that every edge of H occurs in exactly two of the subgraphs and any two of the subgraphs share exactly one edge. Our concept of SOD is a natural generalization of the well-studied orthogonal double covers (ODC) of complete graphs. If for some given G there is an appropriate H, then our goal is to find one with as few vertices as possible. Special attention is paid to the case when G a matching with $n-1$ edges. We conjecture that $v(H)=2n-2$ is best possible if $n\ne 4$ is even and $v(H)=2n$ if n is odd. We present a construction which proves this conjecture for all but 4 of the possible residue classes of n modulo 18.     

On large light graphs in families of polyhedral graphs

A graph H is said to be light in a family H of graphs if each graph G∈H containing a subgraph isomorphic to H contains also an isomorphic copy of H such that each its vertex has the degree (in G) bounded above by a finite number φ(H,H) depending only on H and H. We prove that in the family of all 3-connected plane graphs of minimum degree 5 (or minimum face size 5, respectively), the paths with certain small graphs attached to one of its ends are light.     

Partial characterizations of clique-perfect and coordinated graphs: superclasses of triangle-free graphs

A graph is clique-perfect if the cardinality of a maximum clique-independent set equals the cardinality of a minimum clique-transversal, for all its induced subgraphs. A graph G is coordinated if the chromatic number of the clique graph of H equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The complete lists of minimal forbidden induced subgraphs for the classes of cliqueperfect and coordinated graphs are not known, but some partial characterizations have been obtained. In this paper, we characterize clique-perfect and coordinated graphs by minimal forbidden induced subgraphs when the graph is either paw-free or {gem,W₄,bull}-free, two superclasses of triangle-free graphs.     

Geodesic subgraphs

Define a geodesic subgraph of a graph to be a subgraph H with the property that any geodesic of two points of H is in H. The trivial geodesic subgraphs are the complete graphs K_n' n ≧ 0, and G itself. We characterize all (finite, simple, connected) graphs with only the trivial geodesic subgraphs, and give an algorithm for their construction. We do this also for triangle-free graphs.     

Distance-preserving subgraphs of hypercubes

We give a characterization of connected subgraphs G of hypercubes H such that the distance in G between any two vertices a, b?G is the same as their distance in H. The hypercubes are graphs which generalize the ordinary cube graph.     

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K⁰(G)=G, Kⁿ⁺¹(G)=K(Kⁿ(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that Kⁿ(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G?Kⁿ(G) for all n, moreover Kⁿ(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.     

Hosoya polynomials under gated amalgamations

An induced subgraph H of a graph G is gated if for every vertex x outside H there exists a vertex x^′ inside H such that each vertex y of H is connected with x by a shortest path passing through x^′. The gated amalgam of graphs G₁ and G₂ is obtained from G₁ and G₂ by identifying their isomorphic gated subgraphs H₁ and H₂. Two theorems on Hosoya polynomials of gated amalgams are provided. As their applications, explicit expressions for Hosoya polynomials of hexagonal chains are obtained.     

Matroidal families of finite connected nonhomeomorphic graphs exist

A matroidal family is a nonempty set ? of connected finite graphs such that for every arbitrary finite graph G the edge sets of the subgraphs of G which are isomorphic to an element of ? form a matroid on the edge set of G. In the present paper the question whether there are any matroidal families besides the four previously described by Simões-Pereira is answered affirmatively. It is shown that for every natural number n ? 2 there is a matroidal family that contains the complete graph with n vertices. For n = 4 this settles Simões-Pereira's conjecture that there is a matroidal family containing all wheels.     

Four gravity results

The gravity of a graph H in a given family of graphs H is the greatest integer n with the property that for every integer m, there exists a supergraph G∈H of H such that each subgraph of G, which is isomorphic to H, contains at least n vertices of degree ?m in G. Madaras and Škrekovski introduced this concept and showed that the gravity of the path P_k on k?2 vertices in the family of planar graphs of minimum degree 2 is k-2 for each k≠5,7,8,9. They conjectured that for each of the four excluded cases the gravity is k-3. In this paper we show that this holds.