Linked graphs with restricted lengths

A graph G is k-linked if G has at least 2k vertices, and for every sequence x1, x2, . . . , xk, y1, y2, . . . , yk of distinct vertices, G contains k vertex-disjoint paths P1, P2, . . . , Pk such that Pi joins xi and yi for i = 1, 2, . . . , k. Moreover, the above defined k-linked graphG is k-linked modulo (m1,m2, . . . ,mk) if, in addition, for any ktuple (d1, d2, . . . , dk) of natural numbers, the paths P1, P2, . . . , Pk can be chosen such that Pi has length di modulo mi for i = 1, 2, . . . , k. Thomassen showed that there exists a function f(m1,m2, . . . ,mk) such that every f(m1,m2, . . . ,mk)-connected graph is k-linked modulo (m1,m2, . . . ,mk) provided all mi are odd. For even moduli, he showed in another article that there exists a natural number g(2, 2, · · · , 2) such that every g(2, 2, · · · , 2)-connected graph is k-linked modulo (2, 2, · · · , 2) if deleting any 4k − 3 vertices leaves a non-bipartite graph. In this talk, we show linear upper bounds for f(m1,m2, . . . ,mk) and g(m1,m2, . . . ,mk) in terms of m1,m2, . . . ,mk, respectively. More specifically, we prove the following two results: (i) For any k-tuple (m1,m2, . . . ,mk) of odd positive integers, every max{14(m1+· · ·+mk)−4k, 6(m1+· · ·+mk)−4k+36}-connected graph is k-linked modulo (m1,m2, . . . ,mk). (ii) Let 1 ≤ l ≤ k and let (m1,m2, . . . ,mk) be a k-tuple of positive integers such that mi is odd for each i with l + 1 ≤ i ≤ k. If G is 45(m1 + · · ·+mk)-connected, then either G has a vertex set X of order at most 2k + 2l − 3 + δ(m1, . . . ,ml) such that G − X is bipartite or G is k-linked modulo (2m1, 2m2, . . . , 2ml,ml+1, . . . ,mk), where δ(m1, . . . ,ml) = { 0 if min{m1, . . . ,ml} = 1, and 1 if min{m1, . . . ,ml} ≥ 2. Our results generalize several known results on k-parity-linked graphs. This is a joint work with Guantao Chen, Yuan Chen, Shuhong Gao