Multilevel Bandwidth and Radio Labelings of Graphs

Abstract

This paper introduces a generalization of the graph bandwidth parameter: for a graph G and an integer $k\leq\text{diam}(G)$, the k-level bandwidth B ^k(G) of G is defined by

$B^k(G)=\min_\gamma \max\{|\gamma(x)-\gamma(y)|-d(x,y)+1: x,y\in V(G),d(x,y)\leq k\},$ the minimum being taken among all proper numberings γ of the vertices of G.

We present general bounds on B ^k(G) along with more specific results for k = 2 and the exact value for $k=\text{diam}(G)$. We also exhibit relations between the k-level bandwidth and radio k-labelings of graphs from which we derive a upper bound for the radio number of an arbitrary graph.