@techreport{TR-IC-25-06,
   number = {IC-25-06},
   author = {L. G. S. {Gonzaga} and C. N. Campos},
   title  =  {{Neighbour-distinguishing  edge-labellings  of Powers of
                   Paths}},
   month = {December},
   year = {2025},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 12 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       Given   a   graph   $G$,   the   pair   $(\pi,c_{\pi})$   is  a 
       neighbour-distinguishing          $k$-edge-labelling         if        
       $\pi:E(G)\rightarrow \{1,\ldots,k\}$ such that, for every $v\in
       V(G)$, $c_{\pi}(v) = \sum_{u \in N(v)} \pi(uv)$ and $c_{\pi}(x)
       \neq  c_{\pi}(y)$  for  every edge $xy \in E(G)$. The least $k$
       for  which  it  has  been  shown  that  every  graph  admits  a 
       neighbour-distinguishing   $k$-edge-labelling   is  three.  The 
       $1,2,3$-Conjecture,  proposed  in  2004 by Karo{\'n}ski et al.,
       states   that   every   graph  has  a  neighbour-distinguishing 
       $3$-edge-labelling. This conjecture has been recently proved by
       Keusch  and  published  May  2024.  In  2017,  Luiz  and Campos
       verified   the  $1,2,3$-Conjecture  for  powers  of  paths  and 
       conjectured  that a neighbour-distinguishing $2$-edge-labelling
       could  be  built for powers of paths not isomorphic to complete
       graphs. In this work, we prove this conjecture.
  }
}