Weight Choosability of Graphs

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Alinejad, Mohsen | 2012

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 43863 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saeed
Abstract:
Suppose the edges of a graph G are assigned 3-element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different?
A graph G = (V;E) is called (k; k′)-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f : V [ E ! R such that f(y) 2 L(y) for any y 2 V [ E and for any two adjacent vertices x; x′
Σ e2E(x) f(e) + f(x) ̸=Σ e2E(x′) f(e) + f(x′)
Is it possible every graph is (2, 2)-total weight choosable and every graph without isolated edges is (1, 3)-total weight choosable
Keywords:
Directed Graph ; Permanent Rank ; Weight Choosability