1,393 research outputs found

    The Research on the L(2,1)-labeling problem from Graph theoretic and Graph Algorithmic Approaches

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    The L(2,1) -labeling problem has been extensively researched on many graph classes. In this thesis, we have also studied the problem on some particular classes of graphs. In Chapter 2 we present a new general approach to derive upper bounds for L(2,1)-labeling numbers and applied that approach to derive bounds for the four standard graph products. In Chapter 3 we study the L(2,1)-labeling number of the composition of n graphs. In Chapter 4 we consider the Cartesian sum of graphs and derive, both, lower and upper bounds for their L(2,1)-labeling number. We use two different approaches to derive the upper bounds and both approaches improve previously known bounds. We also present new approximation algorithms for the L(2,1 )-labeling problem on Cartesian sum graphs. In Chapter 5, we characterize d-disk graphs for d\u3e1, and give the first upper bounds on the L(2,1)-labeling number for this class of graphs. In Chapter 6, we compute upper bounds for the L(2,1)-labeling number of total graphs of K_{1,n}-free graphs. In Chapter 7, we study the four standard products of graphs using the adjacency matrix analysis approach. In Chapter 8, we determine the exact value for the L(2,1)-labeling number of a particular class of Mycielski graphs. We also provide, both, lower and upper bounds for the L(2,1)-labeling number of any Mycielski graph

    Multi-triangulations as complexes of star polygons

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    Maximal (k+1)(k+1)-crossing-free graphs on a planar point set in convex position, that is, kk-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at kk-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of kk-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section.Comment: 40 pages, 24 figures; added references, update Section
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