Tuza conjectured that for every graph $G$ the maximum size $\nu$ of a set of edge-disjoint triangles and minimum size $\tau$ of a set of edges meeting all triangles satisfy $\tau \leq 2\nu$. We consider an edge-weighted version of this conjecture, which amounts to packing and covering triangles in multigraphs. Several known results about the original problem are shown to be true in this context, and some are improved. In particular, we answer a question of Krivelevich, who proved that $\tau \leq 2\nu^*$ (where $\nu^*$ is the fractional version of $\nu$) and asked whether this is tight. We prove that $\tau \leq 2\nu^*-\frac{1}{\sqrt{6}}\sqrt{\nu^*}$ and show that this bound is essentially best possible.

Let Fqbe the field of q elements. An (n, k)-affine extractor is a mapping D : Fqn→ {0,1} such that for any k-dimensional affine subspace X ⊆ Fqn, D(x) is an almost unbiased bit when x is chosen uniformly from X. Loosely speaking, the problem of explicitly constructing affine extractors gets harder as q gets smaller and easier as k gets larger. This is reflected in previous results: When q is 'large enough', specifically q = Ω(n2), Gabizon and Raz construct affine extractors for any k ≥ 1. In the 'hardest case', i.e. when q = 2, Bourgain constructs affine extractors for k ≥ δn for any constant (and even slightly subconstant) δ > 0. Our main result is the following: Fix any k ≥ 2 and let d = 5n/k. Then whenever q > 2 · d2and p = char(Fq) > d, we give an explicit (n, k)-affine extractor. For example, when k = δn for constant δ > 0, we get an extractor for a field of constant size Ω((1/δ)2). We also get weaker results for fields of arbitrary characteristic (but can still work with a constant field size when k = δn for constant δ > 0). Thus our result may be viewed as a 'field-size/dimension' tradeoff for affine extractors. For a wide range of k this gives a new result, but even for large k where we do not improve (or even match) the previous result of, we believe that our construction and proof have the advantage of being very simple: Assume n is prime and d is odd, and fix any non-trivial linear map T : Fqn→ Fq. Define QR : Fq→ {0,1} by QR(x) = 1 if and only if x is a quadratic residue. Then, the function D : Fqn→ {0,1} defined by D(x) =△QR(T(xd)) is an (n, k)-affine extractor. Our proof uses a result of Heur, Leung and Xiang giving a lower bound on the dimension of products of subspaces.

A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O ( n log n ) time. The only method previously known for this problem was to compute the globally shortest noncontractible or nonseparating cycle in O (min{ g 3 , n }, n log n ) time, where g is the genus of the surface. As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O ( n log n ) time, a tight octagonal decomposition in O ( gn log n ) time, and a shortest contractible cycle enclosing a nonempty set of faces in O ( n log 2 n ) time.

The unit distance graph $\mathcal{R}$ is the graph with vertex set $\mathbb{R}^2$ in which two vertices (points in the plane) are adjacent if and only if they are at Euclidean distance 1. We prove that the circular chromatic number of $\mathcal{R}$ is at least 4, thus improving the known lower bound of $32/9$ obtained from the fractional chromatic number of $\mathcal{R}$.