The Existence of Subgraphs with Given Properties

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Nahvi, Mina | 2017

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 49853 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saeed
Abstract:
Let G be a graph. We call G an odd graph if all its vertices have odd degrees. Caro conjectured that for every graph G and every integer k, k > 2, there exists a Zk-coloring for the vertices of G so that for all v 2 V (G), the sum of the colors of all vertices in N[v] is not congruent to 0 modulo k. This thesis is mainly devoted to determining a lower bound for the number of vertices of the largest odd induced subgraph of a given graph with no isolated vertices. Another focus of this thesis is to find an upper bound for the number of odd induced subgraphs, as well as odd induced forests, needed to partition V (G), where G is a given graph with even order. At the end, two variations of Caro’s Conjecture are investigated
Keywords:
Odd Subgraph ; Induced Subgraph ; Odd Forest ; Vertex Partitioning ; Caro's Conjecture