Subgraphs Associated with Lists in Graphs and Digraphs

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Ghaffari Baghestani, Afsane | 2017

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 49864 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saeed
Abstract:
Let G be a graph and F : V (G) ...! 2N be a function. The graph G is said to be F-avoiding if there exists an orientation O of G such that d+ O(v) =2 F(v) for every v 2 V (G), where d+O(v) denotes the out-degree of v in the directed graph G with respect to O. In this thesis it is shown that if G is bipartite and jF(v)j dG(v) 2 for every v 2 V (G), then G is F-avoiding. The bound jF(v)j dG(v) 2 is best possible. For every graph G, we conjecture that if jF(v)j dG(v) 1 2 for every v 2 V (G), then G is F-avoiding. We also argue this conjecture; the best possibility and some partial solutions, e.g. for the complete graphs. For graph G, we say the set f(d+ i ; d i ); i = 1; : : : ; ng of non-negative integer pairs, is realizable if there exists an orientation D such that d+ D(vi) = d i ; d D(vi) = d+ i for every vi 2 V (G)
Keywords:
Orientation ; F-Factor ; Directed Graph ; Simple Graph ; Out-Degree