For any even integer k and any integer i, we prove that a （kr ＋i）-regular multigraph contains a k-factor if it contains no more than kr - 3k/2＋ i ＋ 2 cut edges, and this result is the best possible to guarantee the existence of k-factor in terms of the number of cut edges. We further give a characterization for k-factor free regular graphs.
A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a (G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a (G) Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a (G) max{2Δ(G) + 2, Δ(G) + 22} if g(G) 3, a (G) Δ(G) + 2 if g(G) 5, a (G) Δ(G) + 1 if g(G) 7, and a (G) = Δ(G) if g(G) 16 and Δ(G) 3. For series-parallel graphs G, we have a (G) Δ(G) + 1.
