3 problems

Short rainbow circuit conjecture. If $(M,c)$ has no colour-singular elements, then $M$ has a short rainbow circuit. This conjecture extends the corresponding rank-two case from gra…

Let $M$ be a simple regular matroid, let $r(M)$ denote its rank, and let $c$ colour the elements of $M$ with $r(M)+1$ colours, with each colour class containing at least two elemen…

Let $M$ be a simple matroid of rank $n-1$, and let $c$ be a colouring of its ground set $E(M)$ with $n$ colours. A circuit is rainbow when no two of its elements have the same colo…