A duality theorem for graph embeddings

A generalized type of graph covering, called a “Wrapped quasicovering” (wqc) is defined. If K, L are graphs dually embedded in an orientable surface S, then we may lift these embeddings to embeddings of dual graphs K?,L? in orientable surfaces S?, such that S? are branched covers of S and the restrictions of the branched coverings to K?,L? are wqc's of K, L. the theory is applied to obtain genus embeddings of composition graphs G[nK₁] from embeddings of “quotient” graphs G.     

Toroidality of Intersection Graphs of Ideals of Commutative Rings

Let R be a commutative ring with identity and G(R) its intersection graph. In this paper, all toroidal graphs that are intersection graphs are classified. An improvement over the previous results concerning the planarity of these graphs is also presented.     

The Clique Problem in Ray Intersection Graphs

Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochvíl and Ne?et?il (Comment Math Univ Carolinae 31(1):85–93, 1990).     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

Concerning filtrations on commutative rings

We discuss the following three questions concerning multiplicative filtrations on commutative rings: (1) Must every filtration with finite power type be an AP-filtration? (2) Is the intersection of two AP-filtrations always an AP-filtration?, and (3) Does the multiplicity limit for a module of finite type with respect to a O-dimensional filtration on a noetherian ring always exist? We give examples to show that all three questions have in general negative answers for noetherian rings. However, in case of Dedekind domains the questions all have affirmative answers.     

Intersection graphs of curves in the plane

Let V be a set of curves in the plane. The corresponding intersection graph has V as the set of vertices, and two vertices are connected by an edge if and only if the two corresponding curves intersect in the plane.It is shown that the set of intersection graphs of curves in the plane is a proper subset of the set of all undirected graphs. Furthermore, the set of intersection graphs of straight line-segments is a proper subset of the set of the intersection graphs of curves in the plane. Finally, it is shown that for every k ≥ 3, the problem of determining whether an intersection graph of straight line-segments is k-colorable is NP-complete.     

Distance-regular graphs admitting a perfect 1-code

In this paper, we study the problem that which of distance-regular graphs admit a perfect 1-code. Among other results, we characterize distance-regular line graphs which admit a perfect 1-code. Moreover, we characterize all known distance-regular graphs with small valency at most 4, the distance-regular graphs with known putative intersection arrays for valency 5, and all distance-regular graphs with girth 3 and valency 6 or 7 which admit a perfect 1-code.     

Rank‐tolerance graph classes

In this article we introduce certain classes of graphs that generalize ?‐tolerance chain graphs. In a rank‐tolerance representation of a graph, each vertex is assigned two parameters: a rank, which represents the size of that vertex, and a tolerance which represents an allowed extent of conflict with other vertices. Two vertices are adjacent if and only if their joint rank exceeds (or equals) their joint tolerance. This article is concerned with investigating the graph classes that arise from a variety of functions, such as min, max, sum, and prod (product), that may be used as the coupling functions ? and ρ to define the joint tolerance and the joint rank. Our goal is to obtain basic properties of the graph classes from basic properties of the coupling functions. We prove a skew symmetry result that when either ? or ρ is continuous and weakly increasing, the (?,ρ)‐representable graphs equal the complements of the (ρ,?)‐representable graphs. In the case where either ? or ρ is Archimedean or dual Archimedean, the class contains all threshold graphs. We also show that, for min, max, sum, prod (product) and, in fact, for any piecewise polynomial ?, there are infinitely many split graphs which fail to be representable. In the reflexive case (where ? = ρ), we show that if ? is nondecreasing, weakly increasing and associative, the class obtained is precisely the threshold graphs. This extends a result of Jacobson, McMorris, and Mulder [10] for the function min to a much wider class, including max, sum, and prod. We also give results for homogeneous functions, powers of sums, and linear combinations of min and max. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Intersection Dimensions of Graph Classes

The intersection dimension of a graphG with respect to a classA of graphs is the minimumk such thatG is the intersection of at mostk graphs on vertex setV(G) each of which belongs toA. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) ifA is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect toA is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on [Ben-Arroyo Hartman, I., Newman, I., Ziv, R.:On grid intersection graphs, Discrete Math.87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs.     

Mixed unit interval graphs

The class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral.     

Completeness for intersection classes

An intersection representation of a graph is a function gf mapping vertices to sets such that for any u ≠ v, u is adjacent to v iff gf(u) ∩ gf(v) ≠ . The intersection class defined by a set of sets ∑ is the set of all graphs having an intersection representation using sets from ∑. Interval graphs and chordal graphs are well-studied examples of intersection classes.

This paper introduces the notion of completeness for intersection classes. That is, determining precisely what restrictions can be made on the allowable sets without losing the power to represent any graph in the intersection class. The solution to this problem is presented for the chordal graphs.     

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

Random interval graphs

In this paper we introduce a notion ofrandom interval graphs: the intersection graphs of real, compact intervals whose end points are chosen at random. We establish results about the number of edges, degrees, Hamiltonicity, chromatic number and independence number of almost all interval graphs.     

Nonexistence of some Antipodal Distance-regular Graphs of Diameter Four

We find an inequality involving the eigenvalues of a regular graph; equality holds if and only if the graph is strongly regular. We apply this inequality to the first subconstituents of a distance-regular graph and obtain a simple proof of the fundamental bound for distance-regular graphs, discovered by Juri i , Koolen and Terwilliger. Using this we show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. From these results we prove the nonexistence of distance-regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier .     

A common generalization of line graphs and clique graphs

Both the line graph and the clique graph are defined as intersection graphs of certain families of complete subgraphs of a graph. We generalize this concept. By a k-edge of a graph we mean a complete subgraph with k vertices or a clique with fewer than k vertices. The k-edge graph Δ_k(G) of a graph G is defined as the intersection graph of the set of all k-edges of G. The following three problems are investigated for k-edge graphs. The first is the characterization problem. Second, sets of graphs closed under the k-edge graph operator are found. The third problem is the question of convergence: What happens to a graph if we take iterated k-edge graphs?     

On complete intersection toric ideals of graphs

We study the graphs G for which their toric ideals I _G are complete intersections. In particular, we prove that for a connected graph G such that I _G is a complete intersection all of its blocks are bipartite except for at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. In this case, the generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of these graphs satisfy the odd cycle condition. Finally, we characterize all complete intersection toric ideals of graphs which are normal.     

Intersection Theorems for Infinite Families of Convex Sets in Graphs

We prove several Helly-type theorems for infinite families of geodesically convex sets in infinite graphs. That is, we determine the least cardinal n such that any family of (particular) convex sets in some infinite graph has a nonempty intersection whenever each of its subfamilies of cardinality less than n has a nonempty intersection. We obtain some general compactness theorems, and some particular results for pseudo-modular graphs, strongly dismantlable graphs and ball-Helly graphs.     

Feasibility conditions for the existence of walk-regular graphs

A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.     

Characterizations and recognition of circular-arc graphs and subclasses: A survey

Circular graphs are intersection graphs of arcs on a circle. These graphs are reported to have been studied since 1964, and they have been receiving considerable attention since a series of papers by Tucker in the 1970s. Various subclasses of circular-arc graphs have also been studied. Among these are the proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and co-bipartite circular-arc graphs. Several characterizations and recognition algorithms have been formulated for circular-arc graphs and its subclasses. In particular, it should be mentioned that linear time algorithms are known for all these classes of graphs. In the present paper, we survey these characterizations and recognition algorithms, with emphasis on the linear time algorithms.     

A characterization of complete matroid base graphs

This study grew from an attempt to give a local analysis of matroid base graphs. A neighborhood-preserving covering of graphs p:G → H is one such that p restricted to every neighborhood in G is an isomorphism. This concept arises naturally when considering graphs with a prescribed set of local properties. A characterization is given of all connected graphs with two local properties: (a) there is a pair of adjacent points, the intersection of whose neighborhoods does not contain three mutually nonadjacent points; (b) the intersection of the neigh-borhoods of points two apart is a 4-cycle. Such graphs have neighborhoods of the form K_n × K_m for fixed n, m and are either complete matroid base graphs or are their images under neighborhood-preserving coverings. If n ≠ m, the graph is unique; if n = m, there are n ? 3 such images which are nontrivial. These examples prove that no set of properties of bounded diameter can characterize matroid base graphs.