Combinatorial $k$-systoles on a punctured tori and a pairs of pants

In this paper, $S$ denotes a hyperbolic surface homeomorphic to a punctured torus or a pairs of pants. Our interest is the study of emph{textbf{combinatorial $k$-systoles}} that is geodesics with self-intersection number greater than $k$ and with minimal combinatorial length. We show that the maximal intersection number $I_k$ of combinatorial $k$-systoles grows like $k$ and $underset{krightarrow+infty}{limsup}(I_k(S)-k)=+infty$. This answer -- in the case of a pairs of pants and a punctured torus -- a weak version of Erlandsson-Palier conjecture, originally stated for the geometric length.