Some Results and Conjectures on Bounds for Graph Colorings

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Dehghan, Ali | 2011

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 42197 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Mahmoodian, Ebadollah
Abstract:
There are many ways to color the vertices and edges of graphs, such as, rainbow connection, vertex coloring and dynamic coloring. In this thesis, in Chapter 1 we introduce a new coloring, we consider its relationship with some other colorings and we investigate its computational complexity. In chapter 1, we focus on the proper orientation number. The problem of orienting the edges of a given simple graph so that the maximum indegree of vertices is minimized was introduced in 2004. We show that there is a polynomial time algorithm for determining the proper orientation number of a given 3-regular graph. But it is NP-complete to decide if the proper orientation number of a 4-regular graph is 3 or 4. In the next, we show that, it is NP-complete to determine the proper orientation number, for a given planar graph. By the probabilistic method, we find some resulats for graph colorings, where it is not easy to prove these results by classic methods. In the next chapters, we present some new results for the above colorings, by probabilistic methods.
Keywords:
Graph Coloring ; Computational Complexity ; Proper Orientation ; Lovasz Local Lemma