2022/03/23 by Guzmán-Pro, S., Hochstättler, W.
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2203.12549
The first non-trivial case of Hadwiger's conjecture for oriented matroids reads as follows. If O is an M(K4)-free oriented matroid, then O admits a NZ 3-coflow, i.e., it is 3-colourable in the sense of Hochstättler-Nešetřil. The class of gammoids is a class of M(K4)-free orientable matroids and it is the minimal minor-closed class that contains all transversal matroids. Towards proving the previous statement for the class of gammoids, Goddyn, Hochstättler, and Neudauer conjectured that every gammoid has a positive coline (equivalently, a positive double circuit), which implies that all orientations of gammoids are 3-colourable. In this brief note we disprove Goddyn, Hochstättler, and Neudauers' conjecture by exhibiting a large class of bicircular matroids that do not contain positive double circuits.