# Multigraph Factorisation

I have a graph having 6 vertices and its presentation is $$E_{12}^4E_{13}^5E_{14}^6E_{24}^9E_{25}^2E_{35}^9E_{36}L_5^4L_6^{14}$$. This means that there are $$4$$ edges connecting the vertices $$1$$ and $$2$$, there are $$5$$ edges connecting the vertices $$1$$ and $$3$$ and so on and there are $$4$$ loops at the vertex $$5$$ and there are $$14$$ loops at the vertex $$6$$. This is a $$15$$-regular graph if we have the convention that the loops contribute $$1$$ to the degree of a vertex.

It has a $$5$$-regular factor sub-graph having exactly 6 loops, that is $$E_{12}^2E_{13}E_{14}^2E_{24}^3E_{35}^4L_5L_6^5$$. This is just by trial and error method. Is there a theorem which ensures existence of such a factor ?

• So you ask if $nm$-regular multigraph with $nk$ loops always has an $m$-regular sub-multigraph with $k$ loops? – Max Alekseyev May 5 at 20:15