About S-packing coloring of subcubic graphs
A propos de la coloration de S-packing des graphes sous-cubiques
Résumé
For a non-decreasing sequence of integers (a1, a2, . . . , ak ), an (a1, a2, . . . , ak )-packing
coloring of a graph G is a partition of V (G) into k subsets V 1, V 2, . . . , V k such that the
distance between any two distinct vertices x, y ∈ V i is at least a i + 1, 1 ≤ i ≤ k. This paper
studies the packing coloring of subcubic graphs. Gastineau and Togni (2016) [7] asked
whether every subcubic graph except the Petersen graph is (1, 1, 2, 3)-packing colorable.
A subcubic graph G is said to be i-saturated, 0 ≤ i ≤ 2, if every vertex of degree three
in G has at most i neighbors of degree three and it is said to be 3-saturated if it is not
2-saturated. Besides, a vertex of degree three in a 3-saturated subcubic graph is said to be
heavy if all its neighbors are of degree three. We prove that every 1-saturated subcubic
graph is (1, 1, 2)-packing colorable. Moreover, we prove that a 3-saturated subcubic graph
is (1, 1, 2, 2)-packing colorable if any two of its heavy vertices are not adjacent.