The Multi-level Distance Numbers For Two Classes Of Caterpillars

Multi-level distance labeling (or radio labeling) is motivated by the channel assignment problem introduced by Hale. It is a function f:V(G)→{0,1,2,…}, so that the following is satisfied for any u,v∈V(G):|f(u)-f(v)|≥diam(G)+1-d(u,v), where diam(G) is the diameter of G. The span of f is defined as max{|f(u)-f(v)|:u,v∈V(G)}, and the multi-level distance number for a graph G, denoted by rn(G), is the minimum span of all multi-level distance labelings of the graph G.A tree is called a caterpillar if a path or an isolated vertex is obtained by deleting its all pendent vertices and all pendent edges.In this paper, The multi-level distance labeling and the multi-level distance numbers for two classes of caterpillars are studied. One is the particular caterpillar and the other is the caterpillar which is symmetrical about its weight center.According to the contents, this paper is divided into three chapters.In the first chapter, it is introduced that the background, the significance of the paper and the concepts, the present actuality of the multi-level distance labeling and the multi-level distance number at home and abroad, and the preparative theories.In the second chapter, it is studied that a class of particular caterpillars that degree of all vertices are equal, except the pendent vertices. It is obtained that the lower bound by using the proof by contradiction, and then the multi-level distance number is obtained by structuring a particular multi-level labeling.Conclusion 1. Let G= Pkt be a particular caterpillar. We haveIn the third chapter, it is studied that a class of caterpillars which are symmetrical about their weight centers. By using the proof by contradiction and structuring the particular multi-level labeling, we obtain its multi-level distance number.Conclusion 2. Let G= Pkt be a caterpillar which is symmetrical about the weight center. Then...