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arXiv · 1602.03904

On the structure of graphs with given odd girth and large minimum degree

Silvia Messuti·Mathias Schacht
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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 Andr\'asfai, Erd\H os, and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than $\frac{2}{2k+1}n$ must be bipartite. We consider graphs with a weaker condition on the minimum degree. Generalizing results of H\"aggkvist and of H\"aggkvist and Jin for the cases $k=2$ and $3$, we show that every $n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than $\frac{3}{4k}n$ is homomorphic to the cycle of length $2k+1$. This is best possible in the sense that there are graphs with minimum degree $\frac{3}{4k}n$ and odd girth $2k+1$ which are not homomorphic to the cycle of length $2k+1$. Similar results were obtained by Brandt and Ribe-Baumann.

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Silvia Messuti, Mathias Schacht. 2016-02-11. On the structure of graphs with given odd girth and large minimum degree. https://doi.org/10.1002/jgt.21840

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