2 research outputs found
Directed Hamilton cycles in digraphs and matching alternating Hamilton cycles in bipartite graphs
In 1972, Woodall raised the following Ore type condition for directed
Hamilton cycles in digraphs: Let be a digraph. If for every vertex pair
and , where there is no arc from to , we have ,
then has a directed Hamilton cycle. By a correspondence between bipartite
graphs and digraphs, the above result is equivalent to the following result of
Las Vergnas: Let be a balanced bipartite graph. If for any and , where and are nonadjacent, we have , then every perfect matching of is contained in a Hamilton
cycle.
The lower bounds in both results are tight. In this paper, we reduce both
bounds by , and prove that the conclusions still hold, with only a few
exceptional cases that can be clearly characterized.Comment: 16 pages, 7 figures, published on "Siam Journal on Discrete
Mathematics
Spanning Cycles through Specified Edges in Bipartite Graphs
PΓ³sa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgraph of G whose components are nontrivial paths. Let k be the number of edges in F, and let t1 and t2 be the numbers of components of F having odd and even length, respectively. We prove that G has a spanning cycle containing F if any two nonadjacent vertices in opposite partite sets have degree-sum at least n/2 + Ο(F), where Ο(F) = βk/2 β + Ι (here Ι = 1 if t1 = 0 or if (t1, t2) β {(1, 0), (2, 0)}, and Ι = 0 otherwise). We show also that this threshold on the degree-sum is sharp when n> 3k.