William T. Trotter

Extremal problems in discrete geometry

AbstractIn this paper, we establish several theorems involving configurations of points and lines in the Euclidean plane. Our results answer questions and settle conjectures of P. Erdõs, G. Purdy, and G. Dirac. The principal result is that there exists an absolute constantc1 so that when

Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures

\sqrt n \leqq t \leqq \left( {_2^n } \right)

An on-line graph coloring algorithm with sublinear performance ratio

, the number of incidences betweenn points andt lines is less thanc1n2/3t2/3. Using this result, it follows immediately that there exists an absolute constantc2 so that ifk≦√n, then the number of lines containing at leastk points is less thanc2n2/k3. We then prove that there exists an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points. Finally, we show that there is an absolute constantc4 so that there are less than exp (c4 √n) sequences 2≦y1≦y2≦...≦yr for which there is a set ofn points and a setl1,l2, ...,lt oft lines so thatlj containsyj points.

Induced matchings in cubic graphs

Abstract The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G . A conjecture of P. Erdos and S. Burr is settled in the affirmative by proving that for each d ≥ 1, there exists a constant c so that if G is any graph on n vertices with maximum degree d , then the Ramsey number of G is at most cn .

On determinism versus non-determinism and related problems

Abstract A finite, indecomposable partially ordered set is said to be critically indecomposable if, whenever an element is removed, the resulting induced partially ordered set is not indecomposable. The same terminology can be applied to graphs, tournaments, or any other relational structure whose relations are binary and irreflexive. It will be shown in this paper that critically indecomposable partially ordered sets are rather scarce; indeed, there are none of odd order, there is exactly one of order 4, and for each even k ⪖ 6 there are exactly two of order k . The same applies to graphs. For tournaments, there are none have even order, there is exactly one of order 3, and for each odd k ⪖ 5 there are precisely three of order k . In general, for arbitrary irreflexive binary relational structures, we will see that all critical indecomposables fall into one of nine infinite classes. Four of these classes are even—they contain no structures of odd order and for even k ⪖ 6 they each contain (up to a certain type of equivalence) exactly one structure of order k . The five other classes sre odd—they contain no structures of even order and for each odd k ⪖ 5 they each contain exactly one structure of order k . From this characterization of critically indecomposable structures, it will be evident that all indecomposable substructures of critically indecomposable structures are themselves critically indecomposable. Finally, it is proved that every indecomposable structure of order n + 2 ( n ⪖ 5) has an indecomposable substructure of order n .

The maximum number of edges in 2 K 2 -free graphs of bounded degree

In this paper we discuss a generalization of the familiar concept of an interval graph that arises naturally in scheduling and allocation problems. We define the interval number of a graph G to be the smallest positive integer t for which there exists a function f which assigns to each vertex u of G a subset f(u) of the real line so that f(u) is the union of t closed intervals of the real line, and distinct vertices u and v in G are adjacent if and only if f(u) and f(v)meet. We show that (1) the interval number of a tree is at most two, and (2) the complete bipartite graph Km, n has interval number ⌈(mn + 1)/(m + n)⌉.

On the complexity of posets

Abstract One of the simplest heuristics for obtaining a proper coloring of a graph is the First-Fit algorithm: Fix an arbitrary ordering of the vertices and, using the positive integers as the color set, assign to each successive vertex the least integer possible (keeping the coloring proper). This is an example of an on-line algorithm for graph coloring. In the on-line model, a graph is presented one vertex at a time. Each new vertex is given together with all edges joining it to previous vertices. An on-line coloring algorithm assigns a color to each vertex as it is received and once assigned, the color cannot be changed. The performance function , ϱ A ( n ), of an on-line algorithm A is the maximum over all graphs G on n vertices of the ratio of the number of colors used by A to color G to the chromatic numbers of G . The First-Fit algorithm has performance function n /4. We exhibit an algorithm with sublinear performance function.

Explicit matchings in the middle levels of the Boolean lattice

In this paper, we show that the edge set of a cubic graph can always be partitioned into 10 subsets, each of which induces a matching in the graph. This result is a special case of a general conjecture made by Erdos and NeSetiil: For each d 2 3, the edge set of a graph of maximum degree d can always be partitioned into [5d2/4] subsets each of which induces a matching. 0 1993 John Wiley & Sons, Inc.