Partial degrees on matchings for cycles in bipartite graphs

Let G = (V-1 ,V-2 ; E) be a bipartite graph with vertical bar V-1 vertical bar = vertical bar V-2 vertical bar = n . Let M be a matching of G with vertical bar M vertical bar >= 2. Let V(M) denote the set of vertices that are incident with edges of M . We show that if d(x) >= 3n/4 for each x is an element of V(M) and vertical bar M vertical bar > n/4 + 1 with n >= 5 then G contains a cycle covering M and G contains left perpendicular(vertical bar M vertical bar - 1)/2right perpendicular disjoint cycles covering M such that each of the left perpendicular(vertical bar M vertical bar - 1)/2right perpendicular cycles contains at least two edges of M . When vertical bar M vertical bar <= n/4 + 1 or n < 5, the same conclusion holds unless vertical bar M vertical bar is odd and G belongs to one known class of bipartite graphs. We conjecture that if vertical bar M vertical bar > 3n/8, n >= 5 and d(x) >= 3n/4 for each x is an element of V (M) then for any integer partition vertical bar M vertical bar = m(1) + center dot center dot center dot + m (k) with m(i) >= 2 for all i is an element of{1, ..., k}, G contains k disjoint cycles C-1, ..., C-k such that C-i contains m(i) edges of M for all i is an element of{ 1 , ... , k } , unless G belongs to one known class of bipartite graphs. If the conjecture is true, then the lower bound on vertical bar M vertical bar is sharp in general. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.