Nicolas Gastineau

Learn More
An i-packing in a graph G is a set of vertices at pairwise distance greater than i. For a nondecreasing sequence of integers S = (s1, s2,. . .), the S-packing chromatic number of a graph G is the least integer k such that there exists a coloring of G into k colors where each set of vertices colored i, i = 1,. .. , k, is an si-packing. This paper describes(More)
This work establishes the complexity class of several instances of the S-packing coloring problem: for a graph G, a positive integer k and a non decreasing list of integers S = a si-packing (a set of vertices at pairwise distance greater than si). For a list of three integers, a dichotomy between NP-complete problems and polynomial time solvable problems is(More)
Gyárfás et al. and Zaker have proven that the Grundy number of a graph G satisfies Γ(G) ≥ t if and only if G contains an induced subgraph called a t-atom. The family of t-atoms has bounded order and contains a finite number of graphs. In this article, we introduce equivalents of t-atoms for b-coloring and partial Grundy coloring. This concept is used to(More)
such that any two vertices with color s i are at mutual distance greater than s i , 1 ≤ i ≤ k. This paper studies S-packing colorings of (sub)cubic graphs. We prove that subcubic graphs are (1, 2, 2, 2, 2, 2, 2)-packing colorable and (1, 1, 2, 2, 3)-packing colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we provide an example of(More)
Let k ≥ 2 be an integer and T1,. .. , T k be spanning trees of a graph G. If for any pair of vertices (u, v) of V (G), the paths from u to v in each Ti, 1 ≤ i ≤ k, do not contain common edges and common vertices, except the vertices u and v, then T1,. .. , T k are completely independent spanning trees in G. For 2k-regular graphs which are 2k-connected, such(More)
The Grundy number of a graph G, denoted by Γ(G), is the largest k such that there exists a partition of V (G), into k independent sets V1,. .. , V k and every vertex of Vi is adjacent to at least one vertex in Vj , for every j < i. The objects which are studied in this article are families of r-regular graphs such that Γ(G) = r + 1. Using the notion of(More)
The question of whether subcubic graphs have finite packing chromatic number or not is still open although positive responses are known for some subclasses, including subcubic trees, base-3 Sierpiski graphs and hexagonal lattices. In this paper, we answer positively to the question for some subcubic outerplanar graphs. We provide asymptotic bounds depending(More)
The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by defining (i, j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges,(More)
Let k ≥ 2 be an integer and T1,. .. , T k be spanning trees of a graph G. If for any pair of vertices (u, v) of V (G), the paths from u to v in each Ti, 1 ≤ i ≤ k, do not contain common edges and common vertices, except the vertices u and v, then T1,. .. , T k are completely independent spanning trees in G. For 2k-regular graphs which are 2k-connected, such(More)