@techreport{TR-IC-25-05,
   number = {IC-25-05},
   author = {M. M. Omai, C. N. Campos and Atílio G. Luiz},
   title = {{(2,1)-total number of complete equipartite graphs}},
   month = {November},
   year = {2025},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 12 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       We   investigate   the  $(2,1)$-total  labelling  for  complete 
       equipartite   graphs   $K_{r   \times  n}$.  Motivated  by  the 
       conjecture  of  Havet and Yu, which states that every graph $G$
       satisfies   $\lambda_2^t(G)   \leq   \Delta(G)+3$,  we  provide 
       constructive labellings that support this conjecture for almost
       all  cases  of $K_{r \times n}$ with $r \geq 3$ and $n \geq 2$.
       The  only  exception  is  when  $r$ is even and $n$ is odd, for
       which we establish a new upper bound of $\Delta(G) + r + 2$.
  }
}