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

An S-packing k-coloring of a graph \(G\) (with \(S=(s_1,s_2,\dots)\) is a non-decreasing sequence of positive integers) is a mapping \(f\) from \(V(G)\) to \(\lbrace 1,\dots , k \rbrace\) (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 \(\chi_S (G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an S-packing k-coloring. Given a set \(D\subset \mathbb{N}^*\), a distance graph \(G(\mathbb{Z}, D)\) with distance set \(D\) is a graph with vertex set \(\mathbb{Z}\) and two distincts vertices \(u\) and \(v\) are adjacents if \(| u-v | \in D\). In this paper, for \(S=(s,s+1,s+1,\dots)\) with \(s \geq \left\lceil \frac{t}{2} \right\rceil\) we give a lower bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and a lower bound of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) with \(d \geq \left\lceil \frac{t}{2} \right\rceil\), for \(S=(s_1,s_2,\dots, s_i,a,a,\dots)\) with \(a \geq \max( 1 , t-2 )\) we give an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and we determine the exact values of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) and also of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(s\geq \max (\left\lceil \frac{t}{2} \right\rceil, t-3)\) and \(d \geq \max (\left\lceil \frac{t}{2} \right\rceil , t-2)\). And we give a lower and an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(S=(1,s,s,\dots)\) with conditions on \(s\) and \(t\), which in the cases \(s\geq \max (t-2,\left\lceil \frac{t}{2}\right\rceil)\) we determine the exact values of \(\chi_S (G(\mathb b{Z}, \lbrace 1, t\rbrace))\).

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.