We present results on partitioning the vertices of 2-edge-colored graphs
into monochromatic paths and cycles. We prove asymptotically the two-color case
of a conjecture of S\'ark\"ozy: the vertex set of every 2-edge-colored graph
can be partitioned into at most 2α(G) monochromatic cycles, where
α(G) denotes the independence number of G. Another direction, emerged
recently from a conjecture of Schelp, is to consider colorings of graphs with
given minimum degree. We prove that apart from o(∣V(G)∣) vertices, the vertex
set of any 2-edge-colored graph G with minimum degree at least
(1+\eps){3|V(G)|\over 4} can be covered by the vertices of two vertex
disjoint monochromatic cycles of distinct colors. Finally, under the assumption
that G does not contain a fixed bipartite graph H, we show that
in every 2-edge-coloring of G, ∣V(G)∣−c(H) vertices can be covered by two
vertex disjoint paths of different colors, where c(H) is a constant depending
only on H. In particular, we prove that c(C4)=1, which is best possible