9 research outputs found
A quadratic lower bound for subset sums
Let A be a finite nonempty subset of an additive abelian group G, and let
\Sigma(A) denote the set of all group elements representable as a sum of some
subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the
stabilizer of \Sigma(A). Our result implies that \Sigma(A) = Z/nZ for every set
A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by
Erd\H{o}s and Heilbronn for n prime, and by Vu (with a weaker constant) for
general n.Comment: 12 page
Nowhere-zero 8-flows in cyclically 5-edge-connected, flow-admissible signed graphs
In 1983, Bouchet proved that every bidirected graph with a nowhere-zero
integer-flow has a nowhere-zero 216-flow, and conjectured that 216 could be
replaced with 6. This paper shows that for cyclically 5-edge-connected
bidirected graphs that number can be replaced with 8.Comment: 14 page
Induced trees in triangle-free graphs
We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(c √ log n) vertices, where c is a positive constant. The best known upper bound is (2+o(1)) √ n. This partially answers questions of Erdős, Saks, and Sós and of Pultr.
High Girth Cubic Graphs Map to the Clebsch Graph
We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1)