$$L(1,1)$$ L ( 1 , 1 ) -labelling of the direct product of a complete graph and a cycle

An $$L(j,k)$$ L ( j , k ) -labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least $$j$$ j and that of any two vertices of distance $$2$$ 2 is at least $$k$$ k . The minimum span of all $$L(j,k)$$ L ( j , k ) -labelings of $$G$$ G is denoted by $$\lambda _k^j(G)$$ λ k j ( G ) . Lin and Lam (Discret Math 308:3805–3815, 2008) provided an upper bound of $$\lambda _1^2(K_m \times C_n)$$ λ 1 2 ( K m × C n ) when $$K_m \times C_n$$ K m × C n is the direct product of a complete graph $$K_m$$ K m and a cycle $$C_n$$ C n . And they found the exact value of $$\lambda _1^2(K_m \times C_n)$$ λ 1 2 ( K m × C n ) for some $$m$$ m and $$n$$ n . In this paper, we obtain an upper bound and a lower bound of $$\lambda _1^1(K_m \times C_n)$$ λ 1 1 ( K m × C n ) . As a consequence we compute $$\lambda _1^1(K_m \times C_n)$$ λ 1 1 ( K m × C n ) when $$n$$ n is even or $$n\ge 4m+1$$ n ≥ 4 m + 1 .