Linear and cyclic distance-three labellings of trees

TL;DR

This paper investigates distance-based labelling problems on trees, providing bounds, exact values for specific cases, and efficient algorithms for finite trees, extending understanding of complex graph labelling invariants.

Contribution

It introduces sharp bounds and exact values for $L(h,p,p)$ and $C(h,p,p)$ labelling invariants on trees, along with linear-time approximation algorithms for finite trees.

Findings

Sharp bounds on labelling invariants for trees with bounded degree.

Exact values of invariants for specific tree families.

Linear-time algorithms for certain labelling problems.

Abstract

Given a finite or infinite graph $G$ and positive integers $\ell, h_1, h_2, h_3$, an $L(h_1, h_2, h_3)$-labelling of $G$ with span $\ell$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots, \ell\}$ such that, for $i = 1, 2, 3$ and any $u, v \in V(G)$ at distance $i$ in $G$, $|f(u) - f(v)| \geq h_i$. A $C(h_1, h_2, h_3)$-labelling of $G$ with span $\ell$ is defined similarly by requiring $|f(u) - f(v)|_{\ell} \ge h_i$ instead, where $|x|_{\ell} = \min\{|x|, \ell-|x|\}$. The minimum span of an $L(h_1, h_2, h_3)$-labelling, or a $C(h_1, h_2, h_3)$-labelling, of $G$ is denoted by $\lambda_{h_1,h_2,h_3}(G)$, or $\sigma_{h_1,h_2,h_3}(G)$, respectively. Two related invariants, $\lambda^*_{h_1,h_2,h_3}(G)$ and $\sigma^*_{h_1,h_2,h_3}(G)$, are defined similarly by requiring further that for every vertex $u$ there exists an interval $I_u$ $\mod~(\ell + 1)$ or $\mod~\ell$, respectively, such that the neighbours of $u$ are assigned labels from $I_u$ and $I_v \cap I_w = \emptyset$ 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 $h \ge p \ge 1$ are integers. We give sharp bounds on $\lambda_{h,p,p}(T)$, $\lambda^*_{h,p,p}(T)$, $\sigma_{h, 1, 1}(T)$ and $\sigma^*_{h, 1, 1}(T)$, 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 a few families of trees. We give sharp bounds on $\sigma_{h,p,p}(T)$ and $\sigma^*_{h,p,p}(T)$ for trees with maximum degree $\Delta \le h/p$, and as a special case we obtain that $\sigma_{h,1,1}(T) = \sigma^*_{h,1,1}(T) = 2h + \Delta - 1$ for any tree $T$ with $\Delta \le h$.