Fast generation of regular graphs and construction of cages

The construction of complete lists of regular graphs up to isomorphism is one of the oldest problems in constructive combinatorics. In this article an efficient algorithm to generate regular graphs with a given number of vertices and vertex degree is introduced. The method is based on orderly generation refined by criteria to avoid isomorphism checking and combined with a fast test for canonicity. The implementation allows computing even large classes of graphs, like construction of the 4‐regular graphs on 18 vertices and, for the first time, the 5‐regular graphs on 16 vertices. Also in cases with given girth, some remarkable results are obtained. For instance, the 5‐regular graphs with girth 5 and minimal number of vertices were generated in less than 1 h. There exist exactly four (5, 5)‐cages. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 137–146, 1999     

On Homotopy of connected graphs having the same degree function

Let denote the set of simple graphs having the degree function . It is known that any two graphs from are homotopic in the sense that one can be obtained from the other by a series of simple two-edge switches so that every intermediate graph is in . Let denote the set of simple graphs having at most k components. A natural question arises whether it is possible to extend the above homotopy result on the class , and in particular, whether it is possible to obtain a connected simple graph from another connected simple graph of the same degree function by a series of two edge switches preserving connectivity. In this paper we give an answer to this question. We also consider two-edge switches of a special type and single out some classes of graphs for which these special operations provide the corresponding homotopy between its members.     

Linear-time computation of optimal subgraphs of decomposable graphs

A general problem in computational graph theory is that of finding an optimal subgraph H of a given weighted graph G. The matching problem (which is easy) and the traveling salesman problem (which is not) are well-known examples of this general problem. In the literature one can also find a variety of ad hoc algorithms for solving certain special cases in linear time. We suggest a general approach for constructing linear-time algorithms in the case where the graph G is defined by certain rules of composition (as are trees, series-parallel graphs, and outerplanar graphs) and the desired subgraph H satisfies a property that is “regular” with respect to these rules of composition (as do matchings, dominating sets, and independent sets for all the classes just mentioned). This approach is applied to obtain a linear-time algorithm for computing the irredundance number of a tree, a problem for which no polynomial-time algorithm was previously known.     

Probabilistic Inductive Classes of Graphs

A unifying framework—probabilistic inductive classes of graphs (PICGs)—is defined by imposing a probability space on the rules and their left elements from the standard notion of inductive class of graphs. The rules can model the processes creating real-world social networks, such as spread of knowledge, dynamics of acquaintanceships or sexual contacts, and emergence of clusters. We demonstrate the characteristics of PICGs by casting some well-known models of growing networks in this framework. Results regarding expected size and order are derived. For PICG models of connected and 2-connected graphs order, size and asymptotic degree distribution are presented. The approaches used represent analytic alternative to computer simulation, which is mostly used to obtain the properties of evolving graphs.     

Star partitions and the graph isomorphism problem

A canonical basis of Rⁿ associated with a graph G on n vertices has been defined in [15] in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism. We study algorithms for finding the canonical basis and some of its variations. The emphasis is on the following three special cases; graphs with distinct eigenvalues, graphs with bounded eigenvalue multiplicities and strongly regular graphs. We show that the procedure is reduced in some parts to special cases of some well known combinatorial optimization problems, such as the maximal matching problem. the minimal cut problem, the maximal clique problem etc. This technique provides another proof of a result of L. Babai et al. [2] that isomorphism testing for graphs with bounded eigenvalue multiplicities can be performend in a polynomial time. We show that the canonical basis in strongly regular graphs is related to the graph decomposition into two strongly regular induced subgraphs. Examples of distinguishing between cospectral strongly regular graphs by means of the canonical basis are provided. The behaviour of star partitions of regular graphs under operations of complementation and switching is studied.     

The Exact Weighted Independent Set Problem in Perfect Graphs and Related Classes

The exact weighted independent set (EWIS) problem consists in determining whether a given vertex-weighted graph contains an independent set of given weight. This problem is a generalization of two well-known problems, the NP-complete subset sum problem and the strongly NP-hard maximum weight independent set (MWIS) problem. Since the MWIS problem is polynomially solvable for some special graph classes, it is interesting to determine the complexity of this more general EWIS problem for such graph classes.We focus on the class of perfect graphs, which is one of the most general graph classes where the MWIS problem can be solved in polynomial time. It turns out that for certain subclasses of perfect graphs, the EWIS problem is solvable in pseudo-polynomial time, while on some others it remains strongly NP-complete. In particular, we show that the EWIS problem is strongly NP-complete for bipartite graphs of maximum degree three, but solvable in pseudo-polynomial time for cographs, interval graphs and chordal graphs, as well as for some other related graph classes.     

Two-ended regular median graphs

We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends.For cubic median graphs G the condition of linear growth can be weakened to the condition that G has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.     

Site percolation and isoperimetric inequalities for plane graphs

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini, and Horesh that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.     

On Small World Semiplanes with Generalised Schubert Cells

The well known “real-life examples” of small world graphs, including the graph of binary relation: “two persons on the earth know each other” contains cliques, so they have cycles of order 3 and 4. Some problems of Computer Science require explicit construction of regular algebraic graphs with small diameter but without small cycles. The well known examples here are generalised polygons, which are small world algebraic graphs i.e. graphs with the diameter d≤clog  _k−1(v), where v is order, k is the degree and c is the independent constant, semiplanes (regular bipartite graphs without cycles of order 4); graphs that can be homomorphically mapped onto the ordinary polygons. The problem of the existence of regular graphs satisfying these conditions with the degree ≥k and the diameter ≥d for each pair k≥3 and d≥3 is addressed in the paper. This problem is positively solved via the explicit construction. Generalised Schubert cells are defined in the spirit of Gelfand-Macpherson theorem for the Grassmanian. Constructed graph, induced on the generalised largest Schubert cells, is isomorphic to the well-known Wenger’s graph. We prove that the family of edge-transitive q-regular Wenger graphs of order 2q ⁿ , where integer n≥2 and q is prime power, q≥n, q>2 is a family of small world semiplanes. We observe the applications of some classes of small world graphs without small cycles to Cryptography and Coding Theory.     

On Group Choosability of Total Graphs

In this paper, we study the group and list group colorings of total graphs and present group coloring versions of the total and list total colorings conjectures. We establish the group coloring version of the total coloring conjecture for the following classes of graphs: graphs with small maximum degree, two-degenerate graphs, planner graphs with maximum degree at least 11, planner graphs without certain small cycles, outerplanar graphs and near outerplanar graphs with maximum degree at least 4. In addition, the group version of the list total coloring conjecture is established for forests, outerplanar graphs and graphs with maximum degree at most two.     

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

In this paper,we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks Jn,a class of cubic convex polytopes considering the open problem raised in [M.Imran et al.,families of plane graphs with constant metric dimension,Utilitas Math.,in press] and finally Harary graphs H 5,n by partially answering to an open problem proposed in [I.Javaid et al.,Families of regular graphs with constant metric dimension,Utilitas Math.,2012,88:43-57].We prove that these classes of regular graphs have constant metric dimension.     

Some Families of Directed Strongly Regular Graphs Obtained from Certain Finite Incidence Structures

We present many new directed strongly regular graphs by direct construction. We construct these graphs on the collections of antiflags of certain finite incidence structures. In this way, we confirm the feasibility of infinitely many parameter sets that was previously undetermined. We describe some examples of graphs together with their isomorphism classes to demonstrate the fact that our construction method is capable of producing many graphs with same parameters.     

Regular graphs with given girth pair

The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair is proved and simple bounds for their smallest order are developed. Several infinite classes of such graphs are constructed and it is proved that two of these families consist of smallest graphs.     

A New Method for Enumerating Independent Sets of a Fixed Size in General Graphs

We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin [7] holds for all but finitely many graphs. We also use our method to prove special cases of a conjecture of Kahn [13]. In addition, we show that our method is particularly useful for computing the number of independent sets of small sizes in general regular graphs and Moore graphs, and we argue that it can be used in many other cases when dealing with graphs that have numerous structural restrictions.     

Constructing two-weight codes with prescribed groups of automorphisms

We construct new linear two-weight codes over the finite field with q elements. To do so we solve the equivalent problem of finding point sets in the projective geometry with certain intersection properties. These point sets are in bijection to solutions of a Diophantine linear system of equations. To reduce the size of the system of equations we restrict the search for solutions to solutions with special symmetries.Two-weight codes can be used to define strongly regular graphs. We give tables of the two-weight codes and the corresponding strongly regular graphs. In some cases we find new distance-optimal two-weight codes and also new strongly regular graphs.     

Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs

Recursive procedures are obtained for counting isomorphism classes of three-connected graphs and of two-connected graphs without vertices of degree 2. We apply an enumeration tool developed by R. W. Robinson to count non-isomorphic 2-connected graphs: he expressed the sums of cycle indices of automorphism groups of connected graphs in terms of those of their 2-connected components, and we do the same for 2-connected graphs and their 3-connected components.     

Graphs of arbitrary excessive class

We show that there exists a family of r-regular graphs of arbitrarily large excessive index for each integer r greater than 3. Furthermore, we answer a question in Bonisoli and Cariolaro (2007) [1] showing that all the positive integers can be attained as excessive classes of regular graphs.     

On the applications of Extremal Graph Theory to Coding Theory and Cryptography

Explicit constructions in Extremal graph theory give appropriate lower bound for Turan type problems. In the case of prohibited cycles explicit constructions can be used in various problems of Information Security. We observe algebraic constructions of regular graphs of large girth and graphs with large cycle indicator and describe some algorithms of Coding Theory and Cryptography based on such special families of graphs.     

Vertex-magic labelings of regular graphs II

Previously the first author has shown how to construct vertex-magic total labelings (VMTLs) for large families of regular graphs. The construction proceeds by successively adding arbitrary 2-factors to a regular graph of order n which possesses a strong VMTL, to produce a regular graph of the same order but larger size. In this paper, we exploit this construction method. We are able to show that for any r≥4, every r-regular graph of odd order n≤17 has a strong VMTL. We show how to produce strong labelings for some families of 2-regular graphs since these are used as the starting points of our construction. While even-order regular graphs are much harder to deal with, we introduce ‘mirror’ labelings which provide a suitable starting point from which the construction can proceed. We are able to show that several large classes of r-regular graphs of even order (including some Hamiltonian graphs) have VMTLs.     

Hamiltonicity of Amalgams

 An amalgam is obtained from two Halin graphs having skirting cycles of the same length. We are interested in hamiltonicity of amalgams constructed from two identical Halin graphs without any shift along the skirting cycle. We establish hamiltonicity of amalagams constructed from cubic Halin graphs. We give a sufficient condition for hamiltonicity of non-cubic amalgams and characterize infinite classes of non-Hamiltonian amalgams. We also characterize hamiltonicity of amalgams constructed by shifting the component Halin graphs by one and of general amalgams of higher degree. Received: June 23, 1997