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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)

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)

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)

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