5 research outputs found
Small feedback vertex sets in planar digraphs
Let be a directed planar graph on vertices, with no directed cycle of
length less than . We prove that contains a set of vertices
such that has no directed cycle, and if ,
if , and if . This
improves recent results of Golowich and Rolnick.Comment: 5 pages, 1 figure - v3 final versio
Cuts in matchings of 3-connected cubic graphs
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette,
Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and
on even graphs in digraphs whose contraction is strongly connected
(Hochst\"attler). We show that all of them fit into the same framework related
to cuts in matchings. This allows us to find a counterexample to the conjecture
of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all
planar graphs on at most 26 vertices. Finally, we state a new conjecture on
bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio
Four proofs of the directed Brooks' Theorem
We give four new proofs of the directed version of Brook's Theorem and an
NP-completeness result.Comment: 15 page