Monochromatic homeomorphically irreducible trees in 2-edge-colored complete graphs

It has been known that every 2-edge-colored complete graph has a monochromatic connected spanning subgraph. In this paper, we study a condition which can be imposed on such a monochromatic subgraph, and show that almost all 2-edge-colored complete graphs have a monochromatic spanning tree with no vertices of degree 2. As a corollary of our main theorem, we obtain a Ramsey type result: Every 2-edge-colored complete graph of order n >= 8 has a monochromatic tree T with no vertices of degree 2 and vertical bar V(T)vertical bar >= n - 1.