On the Structure of Graphs with Given Odd Girth and Large Minimum Degree |
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| Authors: | Silvia Messuti Mathias Schacht |
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| Affiliation: | 1. FACHBEREICH MATHEMATIK, UNIVERSIT?T HAMBURG, BUNDESSTRA?E 2. 55,, GERMANY;3. 55,, GERMANYContract grant sponsor: Heisenberg‐Programme of the Deutsche Forschungsgemeinschaft. |
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| Abstract: | We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andrásfai, Erd?s, and Sós implies that every n‐vertex graph with odd girth  and minimum degree bigger than  must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of Häggkvist and of Häggkvist and Jin for the cases  and 3, we show that every n‐vertex graph with odd girth  and minimum degree bigger than  is homomorphic to the cycle of length  . This is best possible in the sense that there are graphs with minimum degree  and odd girth  that are not homomorphic to the cycle of length  . Similar results were obtained by Brandt and Ribe‐Baumann. |
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| Keywords: | Andrasfai‐Erdos‐Sos theorem extremal graph theory graph homomorphisms |
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