On 0-Rotatable Graceful Caterpillars

Abstract

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.