Georgia Institute of Technology College of Sciences

In this talk, we discuss our recent progress on the famous directed cycle double cover conjecture of Jaeger. This conjecture asserts that every 2-connected graph admits a collection of cycles such that each edge is in exactly two cycles of the collection. In addition, it must be possible to prescribe an orientation to each cycle so that each edge is traversed in both ways. We plan to define the class of weakly robust trigraphs and prove that a connectivity augmentation conjecture for this class implies general directed cycle double cover conjecture. This is joint work with Martin Loebl.

Consider a graph G and a specified subset A of vertices. An A-path is a path with both ends in A and no internal vertex in A. Gallai showed that there exists a min-max formula for the maximum number of pairwise disjoint A-paths. More recent work has extended this result, considering disjoint A-paths which satisfy various additional properties. We consider the following model. We are given a list of {(s_i, t_i): 0< i < k} of pairs of vertices in A, consider the question of whether there exist many pairwise disjoint A-paths P_1,..., P_t such that for all j, the ends of P_j are equal to s_i and t_i for some value i. This generalizes the disjoint paths problem and is NP-hard if k is not fixed. Thus, we cannot hope for an exact min-max theorem. We further restrict the question, and ask if there either exist t pairwise disjoint such A-paths or alternatively, a bounded set of f(t) vertices intersecting all such paths. In general, there exist examples where no such function f(t) exists; we present an exact characterization of when such a function exists. This is joint work with Daniel Marx.

By the 4-color theorem, every planar graph on n vertices has an independent set of size at least n/4. Finding a simple proof of this fact is a long-standing open problem. Furthermore, no polynomial-time algorithm to decide whether a planar graph has an independent set of size at least (n+1)/4 is known. We study the analogous problem for triangle-free planar graphs. By Grotzsch' theorem, each such graph on n vertices has an independent set of size at least n/3, and this can be easily improved to a tight bound of (n+1)/3. We show that for every k, a triangle-free planar graph of sufficiently large tree-width has an independent set of size at least (n+k)/3, thus giving a polynomial-time algorithm to decide the existence of such a set. Furthermore, we show that there exists a constant c < 3 such that every planar graph of girth at least five has an independent set of size at least n/c.Joint work with Matthias Mnich.

We prove the dcdc conjecture in a class of lean fork graphs, argue that this class is substantial and show a path towards the complete solution. Joint work with Andrea Jimenez.