On the packing number of $ 3 $-token graph of the path graph $ P_n $

Abstract

In 2018, J. M. Gómez et al. showed that the problem of finding the packing number $ \rho(F_2(P_n)) $ of the 2-token graph $ F_2(P_n) $ of the path $ P_n $ of length $ n\ge 2 $ is equivalent to determining the maximum size of a binary code $ S' $ of constant weight $ w = 2 $ that can correct a single adjacent transposition. By determining the exact value of $ \rho(F_2(P_n)) $, they proved a conjecture of Rob Pratt. In this paper, we study a related problem, which consists of determining the packing number $ \rho(F_3(P_n)) $ of the graph $ F_3(P_n) $. This problem corresponds to the Sloane's problem of finding the maximum size of $ S' $ of constant weight $ w = 3 $ that can correct a single adjacent transposition. Since the maximum packing set problem is computationally equivalent to the maximum independent set problem, which is an NP-hard problem, then no polynomial time algorithms are expected to be found. Nevertheless, we compute the exact value of $ \rho(F_3(P_n)) $ for $ n\leq 12 $, and we also present some algorithms that produce a lower bound for $ \rho(F_3(P_n)) $ with $ 13\leq n\leq 44 $. Finally, we establish an upper bound for $ \rho(F_3(P_n)) $ with $ n\geq 13 $.

Keywords

• packing number
• 3-token graphs
• error correcting codes
• binary codes
• algorithms