  A word-representable graph is a simple graph $G$ which can be represented by
a word $w$ over the vertices of $G$ such that any two vertices are adjacent in
$G$ if and only if they alternate in $w$. It is known that the class of
comparability graphs -- the graphs which admit a transitive orientation -- is
precisely the class of graphs that can be represented by a concatenation of
permutations of vertices. The class of bipartite graphs is a subclass of
comparability graphs. While it is an open problem to determine the
representation number of comparability graphs, it was conjectured that the
representation number of bipartite graphs on $n$ vertices is at most $n/4$. In
this paper, we propose a polynomial time relabeling algorithm to produce a word
representing a given bipartite graph which is a concatenation of permutations
of the graph's vertices. Thus we obtain an upper bound for the representation
number of bipartite graphs, which in turn gives us an upper bound for the
dimension of the posets corresponding to bipartite graphs.
