Investigating Odd and Even-Factors of Graphs

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Ansari, Mojtaba | 2020

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 54057 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Akbari, Saeed
Abstract:
In this thesis we investigate different factors of graphs with emphasis of even and odd factors. First, we prove the best existing lower bound for the largest even factor of graphs which is of the size at least 4/7|E(G)|+1 and we show that this bound is tight and classify the graphs which exactly meet this bound[]. Then we present a new lower bound for the largest odd-factor of graphs of even order which is of the size at least 2/5|E(G)|. We prove this bound using a new edge coloring called 1/2-coloring in which edges are colored with blue and red such that in each cycle the number of red edges is bigger than the number of blue ones. We prove in every 1/2-coloring of graphs with δ(G)≥3 or without any bridges, the total number of red edges is bigger than the total number of blue edges which can help us with investigating odd and even factors. We show that this bound is tight too. Then, we discuss some existing open conjectures
Keywords:
Even Factor ; Odd Factor ; Maximum Odd and Even Factor ; Chen and Fan Theorem ; Supereulierian Graphs

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