On sait depuis les travaux de Bricard et de Connelly qu’il existe dans l’espace euclidien des polyèdres (non convexes) qui sont flexibles : on peut les déformer continûment sans changer la forme de leurs faces. La conjecture des soufflets affirme que le volume interieur de ces polyèdres est constant au cours de la déformation. Elle a été démontrée récemment par I. Sabitov, qui a pour cela utilisé des outils algébriques inattendus dans ce contexte.

A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. $S_i$ denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(), on d + 2- m() vertices...

Let $P_k$ be a path on $k$ vertices. In an earlier paper we have proved that each polyhedral map $G$ on any compact $2$-manifold $𝕄$ with Euler characteristic $\chi \left(𝕄\right)\le 0$ contains a path $P_k$ such that each vertex of this path has, in $G$, degree $\le k⌊\frac{5+\sqrt{49-24\chi \left(𝕄\right)}}{2}⌋$. Moreover, this bound is attained for $k=1$ or $k\ge 2$, $k$ even. In this paper we prove that for each odd $k\ge \frac{4}{3}⌊\frac{5+\sqrt{49-24\chi \left(𝕄\right)}}{2}⌋+1$, this bound is the best possible on infinitely many compact $2$-manifolds, but on infinitely many other compact $2$-manifolds the upper bound can be lowered to $⌊(k-\frac{1}{3})\frac{5+\sqrt{49-24\chi \left(𝕄\right)}}{2}⌋$.