Consider $C$, a family of intervals on $R$, such that each $p ∈ R$ is contained in at most $m$ of these intervals. Prove that there is a partition of $C$ into $m$ classes $C_1, ..., C_m$ so that each $C_i$ consists of pairwise disjoint intervals.
My idea is to represent the family of intervals as a bipartite graph, $B$. Then, the problem just becomes finding a partition of the edge set of $B$ into $m$ disjoint matchings, which by König's theorem is equivalent to showing that the maximum degree of a vertex in $B$ is $m$. However, I am at a loss on how to represent this interval graph as a bipartite graph.
NOTE: I'm looking for a graph theoretic way to solve this!