\((s,s+1,s+1,)\)-packing colorings and \((1,s,s,)\)-packing colorings and d-distance colorings of distance graphs \(D(1,t)\)

An S-packing k-coloring of a graph \(G\) (with \(S=(s_1,s_2,)\) is a non-decreasing sequence of positive integers) is a mapping \(f\) from \(V(G)\) to \( 1, , k \) (the set of colors) such that for every two distincts vertices \(x\) and \(y\) in \(V(G)\) with \(f(x)=f(y)=i\) the distance between \(x\) and \(y\) in \(G\) is bigger than \(s_i\). The S-packing chromatic number \(χ_S (G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an S-packing k-coloring. Given a set \(D⊂ N^*\), a distance graph \(G(Z, D)\) with distance set \(D\) is a graph with vertex set \(Z\) and two distincts vertices \(u\) and \(v\) are adjacents if \(| u-v | ∈ D\). In this paper, for \(S=(s,s+1,s+1,)\) with \(s ≥ t/2 \) we give a lower bound of \(χ_S (G(Z, 1, t))\), and a lower bound of \(χ_d (G(Z, 1, t))\) with \(d ≥ t/2 \), for \(S=(s_1,s_2,, s_i,a,a,)\) with \(a ≥ max( 1 , t-2 )\) we give an upper bound of \(χ_S (G(Z, 1, t))\), and we determine the exact values of \(χ_S (G(Z, 1, t))\) and also of \(χ_d (G(Z, 1, t))\) for \(s≥ max ( t/2 , t-3)\) and \(d ≥ max ( t/2 , t-2)\). And we give a lower and an upper bound of \(χ_S (G(Z, 1, t))\) for \(S=(1,s,s,)\) with conditions on \(s\) and \(t\), which in the cases \(s≥ max (t-2, t/2)\) we determine the exact values of \(χ_S (G( b{Z}, 1, t))\).