THE ULAM NUMBER OF INFINITE-GRAPHS

The following is a conjecture of Ulam: In any partition of the integer lattice on the plane into uniformly bounded sets, there exists a set that is adjacent to at least six other sets. Two sets are adjacent if each contain a vertex of the same unit square. This problem is generalized as follows. Given any uniformly bounded partition P of the vertex set of an infinite graph G with finite maximum degree, let P(G) denote the graph obtained by letting each set of the partition be a vertex of P(G) where two vertices of P(G) are adjacent if and only if the corresponding sets have an edge between them. The Ulam number of G is defined as the minimum of the maximum degree of P(G) where the minimum is taken over all uniformly bounded partitions P. We have characterized the graphs with Ulam number 0, 1, and 2. Restricting the partitions of the vertex set to connected subsets, we obtain the connected Ulam number of G. We have evaluated the connected Ulam numbers for several infinite graphs. For instance we have shown that the connected Ulam number is 4 if G is an infinite grid graph. We have settled the Ulam conjecture for the connected case by proving that the connected Ulam number is 6 for an infinite triangular grid graph. The general Ulam conjecture is equivalent to proving that the Ulam number of the infinite triangular grid graph equals 6. We also describe some interesting geometric consequences of the Ulam number, mainly concerning good drawings of infinite graphs.