|
3
1
|
I am looking for a simple degree conditon that ensures the existence of a k-factor in a graph. The k is supposed to be relatively high and I don't mind the condition being a bit strict. Ideally, something of the form $\delta(G) \geq f(k)$. Any suggestions? 10x! To clarify a bit what I'm after: there is a theorem by Nishimura that ensures a k-factor for k not larger than n/4 or so. But I want a k-factor with k approaching n. http://onlinelibrary.wiley.com/doi/10.1002/jgt.3190160205/abstract |
|||||||||
|
|
3
|
A similar theorem is proved in "Relating minimum degree and the existence of a k-factor" by Hartke, Martin and Seacrest. They show that a graph $G$ on $n$ vertices with minimum degree $\delta\geq \frac{n}{2}$ contains a $k$-factor if $kn$ is even and $$k< \frac{\delta+\sqrt{2\delta n-n^2+8}}{2}.$$ Moreover they show that this is optimal up to a small additive constant ($\le 1$). Notice that as $\delta\to n$ we have $k\to n$, as well. |
||||||
|
