2019/07/26 by Wang, Hong
#05C70 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1907.11668
Let D=(V,A) be a directed graph of order n≥ 6. Let W be a subset of V with |W|≥ 6. Suppose that every vertex of W has degree at least (3n-3)/2 in D. Then for any integer partition |W|=n1+n2 with n1≥ 3 and n2≥ 3, D contains two disjoint directed cycles C1 and C2 such that |V(C1)∩ W|=n1 and |V(C2)∩ W|=n2. We conjecture that for any integer partition |W|=n1+n2+⋯ +nk with k≥ 3 and ni≥ 3(1≤ i≤ k), D contains k disjoint directed cycles C1,C2,… , Ck such that |V(Ci)∩ W|=ni for all 1≤ i≤ k. The degree condition is sharp in general.