@techreport{TR-IC-17-14,
   number = {IC-17-14},
   author = {Atilio G. Luiz and C. N. Campos and R. Bruce Richter},
   title  =  {On  $\alpha$-labellings  of  lobsters  and  trees with a
                   perfect matching},
   month = {August},
   year = {2017},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 15 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       A  graceful labelling of a tree $T$ is an injective function $f
       \colon  V(T)  \to \{0,\ldots,|E(T)|\}$ such that $\{|f(u)-f(v)|
       \colon    uv    \in    E(T)\}    =   \{1,\ldots,|E(T)|\}$.   An  
       $\alpha$-labelling  of  a  tree $T$ is a graceful labelling $f$
       with  the  additional  property that there exists an integer $k
       \in  \{0,\ldots,|E(T)|\}$  such  that,  for  each  edge $uv \in
       E(T)$,  either $f(u) \leq k < f(v)$ or $f(v) \leq k < f(u)$. In
       this  work,  we prove that the following families of trees with
       maximum  degree  three  have $\alpha$-labellings: lobsters with
       maximum  degree  three,  without  $Y$-legs and with at most one
       forbidden  ending;  trees  $T$ with a perfect matching $M$ such
       that  the  contraction  $T/M$ has a balanced bipartition and an
       $\alpha$-labelling; and trees with a perfect matching such that
       their  contree  is  a  caterpillar with a balanced bipartition.
       These  results  reinforce  the  conjecture that every tree with
       maximum   degree   three   and   a   perfect  matching  has  an 
       $\alpha$-labelling.
  }
}