  For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path)
that spans all the vertices in the graph. It is known from \cite{muller,garey}
that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal
bipartite graphs. A convex bipartite graph $G$ with bipartition $(X,Y)$ and an
ordering $X=(x_1,\ldots,x_n)$, is a bipartite graph such that for each $y \in
Y$, the neighborhood of $y$ in $X$ appears consecutively. $G$ is said to have
convexity with respect to $X$. Further, convex bipartite graphs are a subclass
of chordal bipartite graphs. In this paper, we present a necessary and
sufficient condition for the existence of a Hamiltonian cycle in convex
bipartite graphs and further we obtain a linear-time algorithm for this graph
class. We also show that Chvatal's necessary condition is sufficient for convex
bipartite graphs. The closely related problem is HAMILTONIAN PATH whose
complexity is open in convex bipartite graphs. We classify the class of convex
bipartite graphs as {\em monotone} and {\em non-monotone} graphs. For monotone
convex bipartite graphs, we present a linear-time algorithm to output a
Hamiltonian path. We believe that these results can be used to obtain
algorithms for Hamiltonian path problem in non-monotone convex bipartite
graphs. It is important to highlight (a) in \cite{keil,esha}, it is incorrectly
claimed that Hamiltonian path problem in convex bipartite graphs is
polynomial-time solvable by referring to \cite{muller} which actually discusses
Hamiltonian cycle (b) the algorithm appeared in \cite{esha} for the longest
path problem (Hamiltonian path problem) in biconvex and convex bipartite graphs
have an error and it does not compute an optimum solution always. We present an
infinite set of counterexamples in support of our claim.
