An injection \(f :V(T) \rightarrow \{0,\ldots ,|E(T)|\}\) of a tree T is a graceful labelling if \(\{|f(u)-f(v)| :uv \in E(T)\}=\{1,\ldots ,|E(T)|\}\). Tree T is 0-rotatable if, for any \(v \in V(T)\), there exists a graceful labelling f of T such that \(f(v)=0\). In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star \(K_{1,s-1}\) to a leaf of a path \(P_n\) with \(n \ge 3\) and \(s \ge \lceil \frac{n}{2} \rceil \); caterpillars with diameter five or six; and caterpillars T with \(\mathrm {diam}(T) \ge 7\) such that, for every non-leaf vertex \(v \in V(T)\), the number of leaves adjacent to v is even and is at least \(2+2((\mathrm {diam}(T)-1)\bmod {2})\). These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable.

如果\（\ {| f（u）-f（）是对树T的注入\（f：V（T）\ rightarrow \ {0，\ ldots，| E（T）| \} \）v）|：uv \ in E（T）\} = \ {1，\ ldots，| E（T）| \} \）。如果对于任何\（v（in V（T）\））存在一个优美的T标记f使得\（f（v）= 0 \），则树T是0旋转的。在这项工作中，以下毛毛虫科被证明是可0旋转的：具有完美匹配的毛毛虫；通过将星形\（K_ {1，s-1} \）的一片叶子链接到路径\（P_n \）的叶子与\（n \ ge 3 \）和\（s \ ge \ lceil \ frac {n} {2} \ rceil \）; 直径为五或六的毛毛虫；和带有\（\ mathrm {diam}（T）\ ge 7 \）的毛毛虫T，使得对于每个非叶子顶点\（v \ in V（T）\），与v相邻的叶子数是偶数，并且至少为\（2 + 2（（\ mathrm {diam}（T）-1）\ bmod {2}）\）。这些结果使人猜想所有直径至少为5的毛毛虫都可以0旋转。