Bounds On The Labelling Numbers Of Chordal Graphs

Motivated by the conjecture on the L(2, 1)-labelling number λ(G) of a graph G by Griggs and Yeh [2] and the question: “Is the upper bound (∆+3)/4 for λ(G) for chordal graphs with maximum degree ∆ is sharp?”, posed by Sakai [3], we study the bounds for λ(G) for chordal graphs in this paper. Let G be a chordal graph on n vertices with maximum degree ∆ and maximum clique number ω. We improve the upper bound (∆+3)/4 on λ(G) and the upper bound (∆ + 2d− 1)/4 on λd(G) with d ≥ 2, answering question of Sakai and improving results of Chang et al. Finally, we study the labelling numbers of r-power paths P r n on n vertices. We obtain λd(P r n) for small integers d ≥ 2 and r ≥ 2, and give a better bound of λd(P r n) for large integers d and r.