Generalized sum graphs

Harary [8] calls a finite, simple graphG asum graph if one can assign to eachv _i V(G) a labelx _i so that{v _i ,v _j }E(G) iffx _i +x _j =x _k for somek. We generalize this notion by replacing x _i +x _j with an arbitrary symmetric polynomialf(x _i ,x _j ). We show that for eachf, not all graphs are f-graphs. Furthermore, we prove that for everyf and every graphG we can transformG into anf-graph via the addition of |E(G)| isolated vertices. This result is nearly best possible in the sense that for allf and for , there is a graphG withn vertices andm edges which, even after the addition ofm-O(n logn) isolated vetices, is not andf-graph.Research supported in part by a U.S.A.-Israel Binational Science Foundation and by a Bergmann Memorial Grant.Research supported in part by the Office of Naval Research contract number N00014-85-K0622.     

The maximum interval number of graphs with given genus

The interval number of a graph G, denoted i(G), is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t compact real intervals. It is known that every planar graph has interval number at most 3 and that this result is best possible. We investigate the maximum value of the interval number for graphs with higher genus and show that the maximum value of the interval number of graphs with genus g is between ?√g? and 3 + ?√3g?. We also show that the maximum arboricity of graphs with genus g is either 1 + ?√3g? or 2 + ?√3g?.     

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.     

Shrinkability of Minimal Elements in Sphere Representations of Posets

Let k be a positive integer and let P = (X,≤) be a poset. We call P a k-sphere order provided there is a mapping f which assigns to each element x $ \in$ X a ball f(x) in Rk so that x ≤ y in P if and only if f(x) $ \subseteq$ f(y). We ask: Given that P is a k-sphere order, does there necessarily exist a representation f with the property that every minimal element of P is assigned a ball of radius zero? The answer is “yes” for k = 1, but “no” for all k ≥ 2.     

On point-halfspace graphs

The following definition is motivated by the study of circle orders and their connections to graphs. A graphs G is called a point-halfspace graph (in R ^k) provided one can assign to each vertex v ? (G) a point p _v R ^k and to each edge e ? E(G) a closed halfspace H_e ? R ^k so that v is incident with e if and only if p _v ? H_e. Let H ^k denote the set of point-halfspace graphs (in R ^k). We give complete forbidden subgraph and structural characterizations of the classes H ^k for every k. Surprisingly, these classes are closed under taking minors and we give forbidden minor characterizations as well. © 1996 John Wiley & Sons, Inc.     

Degrees of freedom versus dimension for containment orders

Given a family of sets L, where the sets in L admit k degrees of freedom, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)nk log n where k is fixed and n is large.Research supported in part by Allon Fellowship and by a grant from Bat Sheva de Rothschild Foundation.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.     

The many faces of circle orders

A finite partially ordered set P is called a circle order if one can assign to each x P a circular disk C _x so that xy iff C _x C _y . It is interesting to observe that many other classes of posets, such as space-time orders, parabola orders, the Loewner order for 2×2 Hermitian matrices, etc. turn out to be exactly circle orders (or their higher dimensional analogues). We give a global proof for these equivalences.Research supported in part by the Office of Naval Research, the Air Force Office of Scientific Research and by DIMACS.     

On the Fractional Intersection Number of a Graph

 An intersection representation of a graph G is a function f:V(G)→2^S (where S is any set) with the property that uv∈E(G) if and only if f(u)∩f(v)≠∅. The size of the representation is |S|. The intersection number of G is the smallest size of an intersection representation of G. The intersection number can be expressed as an integer program, and the value of the linear relaxation of that program gives the fractional intersection number. This is in consonance with fractional versions of other graph invariants such as matching number, chromatic number, edge chromatic number, etc.  We examine cases where the fractional and ordinary intersection numbers are the same (interval and chordal graphs), as well as cases where they are wildly different (complete multipartite graphs). We find the fractional intersection number of almost all graphs by considering random graphs. Received: July 1, 1996 Revised: August 11, 1997     

A note on graphs and sphere orders

A partially ordered set P is called a k-sphere order if one can assign to each element a ∈ P a ball B_a in R^k so that a < b iff B_a ? B_b. To a graph G = (V,E) associate a poset P(G) whose elements are the vertices and edges of G. We have v < e in P(G) exactly when v ∈ V, e ∈ E, and v is an end point of e. We show that P(G) is a 3-sphere order for any graph G. It follows from E. R. Scheinerman [“A Note on Planar Graphs and Circle Orders,” SIAM Journal of Discrete Mathematics, Vol. 4 (1991), pp. 448–451] that the least k for which G embeds in R^k equals the least k for which P(G) is a k-sphere order. For a simplicial complex K one can define P(K) by analogy to P(G) (namely, the face containment order). We prove that for each 2-dimensional simplicial complex K, there exists a k so that P(K) is a k-sphere order. © 1993 John Wiley & Sons, Inc.     

General results on tolerance intersection graphs

In this paper, general results are presented that are related to ?-tolerance intersection graphs previously defined by Jacobson, McMorris, and Mulder. For example, it is shown that all graphs are ?-tolerance intersection graphs for all ?, yet for “nice” ?, almost no graphs are ?-tolerance interval graphs. Additional results about representation of trees are given.