$$L(2,1)$$ L ( 2 , 1 ) -labeling for brick product graphs

Let $$G=(V, E)$$ G = ( V , E ) be a graph. Denote $$d_G(u, v)$$ d G ( u , v ) the distance between two vertices $$u$$ u and $$v$$ v in $$G$$ G . An $$L(2, 1)$$ L ( 2 , 1 ) -labeling of $$G$$ G is a function $$f: V \rightarrow \{0,1,\cdots \}$$ f : V → { 0 , 1 , ⋯ } such that for any two vertices $$u$$ u and $$v$$ v , $$|f(u)-f(v)| \ge 2$$ | f ( u ) - f ( v ) | ≥ 2 if $$d_G(u, v) = 1$$ d G ( u , v ) = 1 and $$|f(u)-f(v)| \ge 1$$ | f ( u ) - f ( v ) | ≥ 1 if $$d_G(u, v) = 2$$ d G ( u , v ) = 2 . The span of $$f$$ f is the difference between the largest and the smallest number in $$f(V)$$ f ( V ) . The $$\lambda $$ λ -number of $$G$$ G , denoted $$\lambda (G)$$ λ ( G ) , is the minimum span over all $$L(2,1 )$$ L ( 2 , 1 ) -labelings of $$G$$ G . In this article, we confirm Conjecture 6.1 stated in X. Li et al. (J Comb Optim 25:716–736, 2013) in the case when (i) $$\ell $$ ℓ is even, or (ii) $$\ell \ge 5$$ ℓ ≥ 5 is odd and $$0 \le r \le 8$$ 0 ≤ r ≤ 8 .