On Packing Colorings of Distance Graphs

The {\em packing chromatic number} $\chi_{\rho}(G)$ of a graph $G$ is the least integer $k$ for which there exists a mapping $f$ from $V(G)$ to $\{1,2,\ldots ,k\}$ such that any two vertices of color $i$ are at distance at least $i+1$. This paper studies the packing chromatic number of infinite distance graphs $G(\mathbb{Z},D)$, i.e. graphs with the set $\mathbb{Z}$ of integers as vertex set, with two distinct vertices $i,j\in \mathbb{Z}$ being adjacent if and only if $|i-j|\in D$. We present lower and upper bounds for $\chi_{\rho}(G(\mathbb{Z},D))$, showing that for finite $D$, the packing chromatic number is finite. Our main result concerns distance graphs with $D=\{1,t\}$ for which we prove some upper bounds on their packing chromatic numbers, the smaller ones being for $t\geq 447$: $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 40$ if $t$ is odd and $\chi_{\rho}(G(\mathbb{Z},\{1,t\}))\leq 81$ if $t$ is even

On interval number in cycle convexity

International audienceRecently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) \ S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S.In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtainthat $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ).On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G.Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems