  The biclique partition number $(\text{bp})$ of a graph $G$ is referred to as
the least number of complete bipartite (biclique) subgraphs that are required
to cover the edges of the graph exactly once. In this paper, we show that the
biclique partition number ($\text{bp}$) of a co-chordal (complementary graph of
chordal) graph $G = (V, E)$ is less than the number of maximal cliques
($\text{mc}$) of its complementary graph: a chordal graph $G^c = (V, E^c)$. We
first provide a general framework of the ``divide and conquer" heuristic of
finding minimum biclique partitions of co-chordal graphs based on clique trees.
Furthermore, a heuristic of complexity $O[|V|(|V|+|E^c|)]$ is proposed by
applying lexicographic breadth-first search to find structures called moplexes.
Either heuristic gives us a biclique partition of $G$ with size
$\text{mc}(G^c)-1$. In addition, we prove that both of our heuristics can solve
the minimum biclique partition problem on $G$ exactly if its complement $G^c$
is chordal and clique vertex irreducible. We also show that $\text{mc}(G^c) - 2
\leq \text{bp}(G) \leq \text{mc}(G^c) - 1$ if $G$ is a split graph.
