Discrete Geometry and Extremal Graph Theory

In 1946, Paul Erdős proposed the following problem: what is the maximum number of unit distances among n points in the plane? Erdős established an upper bound of cn3/2 and a lower bound that grows slightly faster than n. A graph is a collection of “vertices” and “edges” connecting some pairs of vertices. For example, in an airline route chart, airports are the vertices and direct connections correspond to the edges. We examine graphs without a 4-cycle, and show the maximum number of edges to be at most cn3/2. Replacing the 4-cycle by the “Θ-graph,” we obtain an upper bound of c′n3/2 on the number of edges. Finally, we use this seemingly unrelated graph theoretic result to establish Erdős’ upper bound on the number of unit distances. Although we present the optimal solution, within a constant factor, to the graph theory problem, Erdős’ question on unit distances remains wide open after sixty years. A cycle (also known as a circuit) is a subset of the edge set of a graph G that forms a path such that the first node of the path corresponds to the last. A 4-cycle is merely a cycle with 4 vertices, as shown below. Create PDF with GO2PDF for free, if you wish to remove this line, click here to buy Virtual PDF Printer 1 We begin by proving the following. Theorem 1. A graph G on n vertices without a 4-cycle contains at most cn edges for some constant c > 0. Proof of Theorem 1. Let d1, d2, . . . , dn be the degrees of the vertices v1, v2, . . . , vn of G. Let E be the number of edges of G, so 2E = ∑n i=1 di. For a pair of vertices v1 and v2, note that there can be at most one other vertex v3 connected to both v1 and v2. Otherwise, for if v4 was also joined to v1 and v2, we would have a 4-cycle with vertices v1, v3, v2, v4. Thus, more generally, for a vertex vi connected to di other vertices, we count every pair of these di vertices. If we repeat this count for all the vertices v1, v2, . . . , vn, note that our total count must less than the total number of pairs of vertices, ( n 2 ) . This is because every pair of some di vertices connected to vi can be uniquely associated with vi. In other words, we have the following inequality. ( d1 2 ) + ( d2 2 ) + · · · ( dn 2 ) ≤ ( n 2 ) . We claim that ( d1 2 ) + ( d2 2 ) + · · · ( dn 2 ) ≥ 2E2 n − E, so it will follow that 2E n − E ≤ ( n 2 ) = n(n− 1) 2 ⇒ 4E − 2En ≤ n(n− 1). Thus, ( 2E − n 2 )2 ≤ n(n− 1) + n2 4 ≤ n, so we have 2E ≤ n + n 2 . Finally, we conclude that E < ∼ n (and actually, c ≈ 1 2 will suffice for sufficiently large n). Now, we give two proofs of the inequality ( d1 2 ) + ( d2 2 ) + · · ·+ ( dn 2 ) ≥ 2E2 n − E. • First Proof. By the Quadratic-Arithmetic Mean Inequality (or Cauchy-Schwartz), we have( d1 2 ) + ( d2 2 ) + · · ·+ ( dn 2 ) = 1 2 (d1 + d 2 2 + · · ·+ dn)− 1 2 (d1 + d2 + · · ·+ dn) ≥ 1 2n (2E) − E. • Second Proof. As a function, f(x) = ( x 2 ) = x(x−1) 2 is convex, so by Jensen’s inequality, we have ( d1 2 ) + ( d2 2 ) + · · ·+ ( dn 2 ) ≥ n ( 2E n 2 ) = 2E n − E. This concludes our proof. The “Θ-graph” is the union of three internally disjoint (simple) paths that have the same two end vertices, as shown below.