Packing Coloring of Undirected and Oriented Generalized Theta Graphs

Daouya Laïche (1) , Isma Bouchemakh (1) , Eric Sopena (2)

1 L'IFORCE - L'IFORCE
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

The packing chromatic number χ ρ (G) of an undirected (resp. oriented) graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1 ,... , V k , in such a way that every two distinct vertices in V i are at distance (resp. directed distance) greater than i in G for every i, 1 ≤ i ≤ k. The generalized theta graph Θ ℓ 1 ,...,ℓp consists in two end-vertices joined by p ≥ 2 internally vertex-disjoint paths with respective lengths 1 ≤ ℓ 1 ≤ . . . ≤ ℓ p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n 3 = |{i / 1 ≤ i ≤ p, ℓ i = 3}|, and that both these bounds are tight. We then characterize undirected generalized theta graphs with packing chromatic number k for every k ≥ 3. We also prove that the packing chromatic number of any oriented generalized theta graph lies between 2 and 5 and that both these bounds are tight.

Mots clés: Packing coloring; Packing chromatic number; Theta graph; Generalized theta graph

Domaines: Mathématique discrète [cs.DM]

Daouya Laïche, Isma Bouchemakh, Eric Sopena. Packing Coloring of Undirected and Oriented Generalized Theta Graphs. The Australasian Journal of Combinatorics, 2016, 66 (2), pp.310-329. ⟨hal-01326202v2⟩