# Nearly spanning regular subgraphs

\begin{conjecture} For every $\epsilon > 0$ and every positive integer $k$, there exists $r_0 = r_0(\epsilon,k)$ so that every simple $r$-regular graph $G$ with $r \ge r_0$ has a $k$-regular subgraph $H$ with $|V(H)| \ge (1- \epsilon) |V(G)|$. \end{conjecture}

Petersen's theorem asserts that every regular graph of even degree contains a 2-factor (i.e. a spanning 2-regular subgraph). Iterating this easy result we find that for any pair of positive even integers $k,r$, every $r$-regular graph has a spanning $k$-regular subgraph. The cases when either $k$ or $r$ is odd are considerably more complicated. There are some nice general results (see [AFK]) which show that every regular graph of sufficiently high degree contains a $k$-regular subgraph. However these theorems give no bound on the size of this subgraph.

For $k=1$ this conjecture is an easy consequence of Vizing's Theorem. Indeed, this theorem implies that every $d$-regular graph $G$ has a 1-regular subgraph $H$ with $|V(H)| \ge (1 - \frac{1}{d+1}) |V(G)|$ (just choose a largest color class from a $(d+1)$-edge coloring). Alon [A] proved the conjecture for $k=2$ with the help of two famous results on permanents: the Minc Conjecture (proved by Bregman), and the van der Waerden conjecture (proved by Falikman and Egorichev). It is open for all $k \ge 3$.

## Bibliography

*[A] N. Alon, \href[Problems and results in extremal combinatorics]{http://www.math.tau.ac.il/~nogaa/PDFS/extremal1.pdf}, J, Discrete Math. 273 (2003), 31-53.

[AFK] N. Alon, S. Friedland and G. Kalai, Regular subgraphs of almost regular graphs, J. Combinatorial Theory, Ser. B 37(1984), 79-91.

* indicates original appearance(s) of problem.