Linear and cyclic distance-three labellings of trees

Given a finite or infinite graph G and positive integers `, h1, h2, h3, an L(h1, h2, h3)labelling of G with span ` is a mapping f : V (G) → {0, 1, 2, . . . , `} such that, for i = 1, 2, 3 and any u, v ∈ V (G) at distance i in G, |f(u)−f(v)| ≥ hi. A C(h1, h2, h3)-labelling of G with span ` is defined similarly by requiring |f(u)− f(v)|` ≥ hi instead, where |x|` = min{|x|, `− |x|}. The minimum span of an L(h1, h2, h3)-labelling, or a C(h1, h2, h3)-labelling, of G is denoted by λh1,h2,h3(G), or σh1,h2,h3(G), respectively. Two related invariants, λ ∗ h1,h2,h3 (G) and σ∗ h1,h2,h3(G), are defined similarly by requiring further that for every vertex u there exists an interval Iu mod (` + 1) or mod `, respectively, such that the neighbours of u are assigned labels from Iu and Iv ∩ Iw = ∅ for every edge vw of G. A recent result asserts that the L(2, 1, 1)-labelling problem is NP-complete even for the class of trees. In this paper we study the L(h, p, p) and C(h, p, p) labelling problems for finite or infinite trees T with finite maximum degree, where h ≥ p ≥ 1 are integers. We give sharp bounds on λh,p,p(T ), λ ∗ h,p,p(T ), σh,1,1(T ) and σ ∗ h,1,1(T ), together with linear time approximation algorithms for the L(h, p, p)-labelling and the C(h, 1, 1)-labelling problems for finite trees. We obtain the precise values of these four invariants for complete m-ary trees with height at least 4, the infinite complete m-ary tree, and the infinite (m + 1)-regular tree and its finite subtrees induced by vertices up to a given level. We give sharp bounds on σh,p,p(T ) and σ∗ h,p,p(T ) for trees with maximum degree ∆ ≤ h/p, and as a special case we obtain that σh,1,1(T ) = σ ∗ h,1,1(T ) = 2h+ ∆− 1 for any tree T with ∆ ≤ h.