Jack M. Robertson

A characterization and hereditary properties for partition graphs

Abstract A general partition graph is an intersection graph G on a set S so that for every maximal independent set M of vertices in G , the subsets assigned to the vertices in M partition S . It is shown that such graphs are characterized by there existing a clique cover of G so that every maximal independent set has a vertex from each clique in the cover. A process is described showing how to construct a partition graph with certain prescribed graph theoretic invariants starting with an arbitrary graph with similar invariants. Hereditary properties for various graph products are examined. It is also shown that partition graphs are preserved by removal of any vertex whose closed neighborhood properly contains the closed neighborhood of some other vertex.

Approximating fair division with a limited number of cuts

A large class A of finite algorithms for fairly dividing a cake using k of fewer cuts is described. Assume an algorithm assigns piece Xi to player Pi using associated probability measure μi on measurable subsets of the cake X. If M(n, k) = maxA mini(μi(Xi)) and N(n, k) = maxA(number of i such that μ1(X1⩾ 1n) then for n ⩾ 2, M(n, n − 1) = 1(2n − 2), for n ⩾ 3, M(n, n) ⩾ 1(2n − 3), and for n ⩾ 4, M(n, n + 1) ⩾ 1(2n − 4). Also N(2n − 2, n − 1) = n.

Recent examples in the theory of partition graphs

Abstract A partition graph is an intersection graph for a collection of subsets of auniversal set S with the property that every maximal independent set of vertices corresponds to a partition of S . Two questions which arose in the study of partition graphs are answered by recently discovered examples. An enumeration of the partition graphs on ten or fewer vertices is provided.

Graphs associated with triangulations of lattice polygons

Two graphs, the edge crossing graph E and the triangle graph T are associated with a simple lattice polygon. The maximal independent sets of vertices of E and T correspond to the triangulations of the polygon into fundamental triangles. Properties of E and T are derived including a formula for the size of the maximal independent sets in E and T. It is shown that T is a factor graph of edge-disjoint 4-cycles, which gives corresponding geometric information, and is a partition graph as recently defined by the authors and F. Harary. 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): 05 C 99, 51 M 05, 52 A 43.

Permutations associated with billiard paths

Associated with a smooth closed convex curve C, a point P on C, and a natural number n>=3, is a billiard graph whose vertices are permutations on the set {1,2,...,n}. The graph is constructed and applied to billiard properties of C. Recursion properties of the graph as n increases are described with the aid of an appropriate generating function.

Sets associated with the farthest point problem

LetS be a convex compact set in a normed linear spaceX. For each cardinal numbern, defineSn = {x εX:x has exactlyn farthest points inS} andTn = ∪k≥nSk. It is shown that ifX =E thenT3 is countable andT2 is contractible to a point. Properties of associated level curves are given.