@techreport{TR-IC-17-12,
   number = {IC-17-12},
   author = {Atilio G. Luiz and C. N. Campos and R. Bruce Richter},
   title = {{Some families of 0-rotatable graceful caterpillars}},
   month = {August},
   year = {2017},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 17 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       A  graceful labelling of a tree $T$ is an injective function $f
       \colon V(T) \to \{0,1,\ldots,|E(T)|\}$ such that $\{|f(u)-f(v)|
       \colon  uv  \in  E(T)\} = \{1,2,\ldots,|E(T)|\}$. A tree $T$ is
       said to be 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,  it is proved that the following families of caterpillars
       are   0-rotatable:   caterpillars   with  a  perfect  matching; 
       caterpillars obtained by identifying a central vertex of a path
       $P_n$  with a vertex of $K_2$; caterpillars obtained by linking
       one leaf of the star $K_{1,s-1}$ to a leaf of a path $P_n$ with
       $n  \geq  3$  and  $s  \geq  \lceil  \frac{n}{2}  \rceil$;  and 
       caterpillars with diameter five or six. These results reinforce
       the  conjecture  that  all  caterpillars with diameter at least
       five are 0-rotatable.
  }
}