General Theory: Hypergraphs. Fractional Matching. Fractional Coloring. Fractional Edge Coloring. Fractional Arboricity and Matroid Methods. Fractional Isomorphism. Fractional Odds and Ends. Appendix. Bibliography. Indexes.

A Richman game is a combinatorial game in which, rather than alternating moves, the two players bid for the privilege of making the next move. We consider both the case where the players pay each other and the case where the players pay a neutral third party. We find optimal strategies considering both the case where the players know how much money their opponent has and the case where they do not.

The most familiar construction of graphs whose clique number is much smaller than their chromatic number is due to Mycielski, who constructed a sequence G(n) of triangle-free graphs with chi(G(n)) = n. In this article, we calculate the fractional chromatic number of G, and show that this sequence of numbers satisfies the unexpected recurrence a(n+1) = a(n) + (1/a(n)). (C) 1995 John Wiley & Sons, Inc.

A natural product on integer programming problems with nonnegative coefficients is defined. Hypergraph covering problems are a special case of such integer programs, and the product defined is a generalization of the usual hypergraph product. The main theorem of this paper gives a sufficient condition under which the solution to the nth power of an integer program is asymptotically as good as the solution to the same nth power when the variables are not necessarily integral but may be arbitrary nonnegative real numbers.

On alternate days, each of two players eats a cookie from a cookie jar. Every cookie has a spoilage date after which it can not be eaten. The object of the game is to eat the last edible cookie. In this paper, we produce a strategy for winning this game when the cookies have distinct spoilage dates.

Place arbitrary integers on the four corners of a square. Then place on the midpoint of each side of the square the absolute value of the difference of the numbers associated with the adjacent corners. Connect the midpoints of the sides of the square to form a new square with integers on its corners. Now repeat the process. FIGURE 1 shows an example starting with the numbers 1, 2, 4, and 7. Much has been written about this process ([1], [3], [4], [5], [6], [7])-the so-called four-numbers game-and its generalizations [8]. The earliest published reference seems to be in [2], where it is attributed to E. Ducci of Italy. It is a simple exercise to show that upon iteration the procedure eventually produces a square of zeroes. What is surprising is how fast this convergence actually happens in practice. Ask someone to pick four numbers play the game, and you will probably find it takes eight or fewer iterations to converge to the zero square. This is despite the fact that the convergence time is unbounded-a slightly harder exercise. It is the purpose of this note to calculate the distribution of convergence times with respect to the natural probability measure on labeled squares and thereby to explain the surprising speed of convergence. Let us generalize the process slightly by permitting real numbers on the corners of the squares. For convenience, we formulate the problem as follows. Let T: R4 R4 be defined by T(a, b, c, d) = (la-bl, lb-cl, Ic-dl, Id-al). For vtE R4, let the convergence time of the four-numbers game starting v be n(vG) = min{m: m > 0 and Tm(vt) = (0, 0, 0, 0)). We wish to calculate the probability that n(v ) = k for small natural numbers k. Since we have no way of making sense of choosing an integer or a real number at random, let us assume that the numbers are chosen according to a uniform distribution on [0, N] for some very large N. (In fact, the distribution of the function n is easily seen to be independent of the choice of N.)

The fractional chromatic number of a graph G lies between the clique number and the chromatic number of G. In many familiar examples the fractional chromatic number is within one unit of the clique number. Here we show that this is not always the case by constructing a sequence of graphs with clique number 2 and with fractional chromatic number given by the sequence (an) with a0 = 1 and an+1 = an + (1/an).