Deriving Bounds on $L(k_1,k_2)$ Labeling of Triangular Lattice by Exploring Underlined Graph Structures

Abstract

The $L(k_1,k_2)$ labeling is a type of labeling of vertices of a graph G(V, E) where absolute difference between labels of vertices at graph distances 1 and 2 are at least $k_1$ and $k_2$, respectively, where $k_1$ and $k_2$ are two non-negative real numbers. The objective of $L(k_1,k_2)$ labeling problem is to find a labeling f of the vertices of G such that the span is minimized, where span $\lambda (G;f)$ of a labeling f is defined as $\displaystyle \max _{u \in V} f(u) - \min _{v \in V} f(v)$. The motivation for $L(k_1,k_2)$ labeling problem for infinite triangular grid $T_\varDelta$ comes from the fact that a class of frequency assignment problems in cellular networks can be modeled as $L(k_1,k_2)$ labeling problems of $T_\varDelta$. Existing bounds on $\lambda (T_\varDelta ;f)$ for $k_1 \le k_2$ are based on enumerations through partial computer simulations. In this paper, we attempt to derive $\lambda (T_\varDelta ;f)$ theoretically by exploring the underlined graph structures without any computer simulations. Specifically, we first select a sub graph of $T_\varDelta$ which holds certain structural properties and then using these properties, we find the lower bound of span of the concerned subgraph. This finally leads us to obtain the lower bound of $\lambda (T_\varDelta ;f)$. We establish that $\lambda (T_\varDelta ;f) \ge 3+2h$ when $h < 1/2$ and $\lambda (T_\varDelta ;f) \ge 4$ when $h \ge 1/2$, where $h=k_1/k_2$. Ours is the first attempt to derive $\lambda (T_\varDelta ;f)$ theoretically and our obtained results exactly coincide with that of the known bound obtained through computer simulations when $0 \le h \le 1/3$. For $h > 1/3$, such known bounds are actually finer than ours.