Further results and questions on S-packing coloring of subcubic graphs

• Published in: Discrete mathematics Vol. 348; no. 4; p. 114376
• Main Authors: Mortada, Maidoun, Togni, Olivier
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2025; Elsevier
• Subjects: Coloring; Combinatorics; Computer Science; Cubic graph; Discrete Mathematics; Graph; Mathematics; Packing chromatic number; S-packing coloring; Saturated subcubic graph; Coloring; Packing chromatic number; Graph; S-packing coloring; Cubic graph; Saturated subcubic graph
• ISSN: 0012-365X
• DOI: 10.1016/j.disc.2024.114376

For a non-decreasing sequence of integers S=(a1,a2,…,ak), an S-packing coloring of G is a partition of V(G) into k subsets V1,V2,…,Vk such that the distance between any two distinct vertices x,y∈Vi is at least ai+1, 1≤i≤k. We consider the S-packing coloring problem on subclasses of subcubic graphs: For 0≤i≤3, a subcubic graph G is said to be i-saturated if every vertex of degree 3 is adjacent to at most i vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and G is said to be (3,i)-saturated if every heavy vertex is adjacent to at most i heavy vertices. We prove that every 1-saturated subcubic graph is (1,1,3,3)-packing colorable and (1,2,2,2,2)-packing colorable. We also prove that every (3,0)-saturated subcubic graph is (1,2,2,2,2,2)-packing colorable.