Manley Perkel

Representations of graphs modulo n

A graph is representable modulo n if its vertices can be labeled with distinct integers between 0 and n, the difference of the labels of two vertices being relatively prime to n if and only if the vertices are adjacent. Erdős and Evans recently proved that every graph is representable modulo some positive integer. We derive a combinatorial formulation of representability modulo n and use it to characterize those graphs representable modulo certain types of integers, in particular integers with only two prime divisors. Other facets of representability are also explored. We obtain information about the values of n modulo which paths and cycles are representable.

Expansion of layouts of complete binary trees into grids

Let Th be the complete binary tree of height h. Let M be the infinite grid graph with vertex set Z2, where two vertices (x1, y1) and (x2, y2) of M are adjacent if and only if |x1-x2| + |y1-y2|= 1. Suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansion ratio n(T)/n(Th) = n(T)/(2h+1 - 1) is bounded below by 1.122 for h sufficiently large. That is, we give bounds on how many vertices of degree 2 must be inserted along the edges of Th in order that the resulting tree can be laid out in the grid. Concerning the constructive end of VLSI design, suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of the n × n grid graph. Define the expansion ratio of such a layout to be n2/n(Th)=n2/(2h+1 - 1). We show constructively that the minimum possible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h. That is, we give efficient layouts of complete binary trees into square grids, making improvements upon the previous work of others. We also give bounds for the point expansion and expansion problems for layouts of Th into extended grids, i.e. grids with added diagonals.

The steiner problem in the hypercube

Let Q(n) be the n-dimensional hypercube, and X, a set of points in Q(n). The Steiner problem for the hypercube is to find the smallest possible number L(n,X) of edges in any subtree of Q(n) that spans X. We obtain the following results: 1 An exact formula for L(n,X), when |X| ≤ 5. 2 The bound L(n,X) ≤ (nk+1) + (2 + o(1)) ([log (k)]/k)(nk) as k ∞, when X is the set of all points in Q(n) of a given weight k + 1, provided (k2/[log (k)])1 + 1/k ≤ n. 3 NP-completeness of deciding L(n,X) even when every point of X has weight at most 2.

Groups of typeF a,b,−c

For integersa, b andc, the groupFa,b,−c is defined to be the group 〈R, S : R2=RSaRSbRS−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of orderpn (for certainp andn) as groups of typeFa,b,−c for certain (not unique) choices ofa, b andc.

On Narrow Hexagonal Graphs with a 3-Homogeneous Suborbit

A connected graph of girth m ≥ 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size k − 1 at distance three from a given vertex.

Arrangements of k-sets with intersection constraints

Given positive integers n,k where n>=k, let f(n,k) denote the largest integer s such that there exists a cyclic ordering of the k-sets on [n]={0,1,...,n-1} such that every s consecutive k-sets are pairwise intersecting. Equivalently, f(n,k) is the largest s such that the complement K(n,k)@? of the Kneser graph K(n,k) contains the sth power of a Hamiltonian cycle. For each n>=6 we show that f(n,2)=3. We show that f(n,3) equals either 2n-8 or 2n-7 when n is sufficiently large, conjecturing that 2n-8 is the correct value. For each k>=4 and n sufficiently large we show that 2n^k^-^2(k-2)!-(72k-2)n^k^-^3(k-3)!-O(n^k^-^4)@?f(n,k)@?2n^k^-^2(k-2)!-(72k-3.2)n^k^-^3(k-3)!+o(n^k^-^3).

Packing an Equilateral Polygon in a Thin Strip

AbstractLetPm denote an equilateral polygon ofm sides with each side having length 1 and we allow the sides to cross and vertex repetitions. We consider the following question. What is the smallest widthtm of a horizontal strip in the Euclidean plane that contains aPm? This problem has its origins in Euclidean Ramsey theory. Whenm is even, it is easy to see thattm=0. For a polygon with an odd number of sides, we prove that

A characterization of J1 in terms of its geometry

\begin{gathered} t_{2n + 1} = \frac{{\sqrt {2n + 1} }}{{n + 1}} for 2n + 1 \equiv 3(\bmod 4) and \hfill \\ t_{2n + 1} = \sqrt {\frac{{2n + 1}}{{n^2 + 2n}}} for 2n + 1 \equiv 1(\bmod 4) \hfill \\ \end{gathered}