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,…,ykx1,x2,…,xk,y1,y2,…,yk of distinct vertices, G contains k   vertex-disjoint paths P1,P2,…,PkP1,P2,…,Pk such that PiPi joins xixi and yiyi for i=1,2,…,ki=1,2,…,k. Moreover, the above defined k-linked graph G is modulo  (m1,m2,…,mk)(m1,m2,…,mk)-linked if, in addition, for any k  -tuple (d1,d2,…,dk)(d1,d2,…,dk) of natural numbers, the paths P1,P2,…,PkP1,P2,…,Pk can be chosen such that PiPi has length didi modulo mimi for i=1,2,…,ki=1,2,…,k. Thomassen showed that, for each k  -tuple (m1,m2,…,mk)(m1,m2,…,mk) of odd   positive integers, there exists a natural number f(m1,m2,…,mk)f(m1,m2,…,mk) such that every f(m1,m2,…,mk)f(m1,m2,…,mk)-connected graph is modulo  (m1,m2,…,mk)(m1,m2,…,mk)-linked  . For m1=m2=…=mk=2m1=m2=…=mk=2, he showed in another article that there exists a natural number g(2,k)g(2,k) such that every g(2,k)g(2,k)-connected graph G is modulo  (2,2,…,2)(2,2,…,2)-linked   or there is X⊆V(G)X⊆V(G) such that |X|⩽4k−3|X|⩽4k−3 and G−XG−X is a bipartite graph, where (2,2,…,2)(2,2,…,2) is a k-tuple.We showed that f(m1,m2,…,mk)⩽max{14(m1+m2+⋯+mk)−4k,6(m1+m2+⋯+mk)−4k+36}f(m1,m2,…,mk)⩽max{14(m1+m2+⋯+mk)−4k,6(m1+m2+⋯+mk)−4k+36} for every k  -tuple of odd positive integers. We then extend the result to allow some mimi be even integers. Let (m1,m2,…,mk)(m1,m2,…,mk) be a k  -tuple of natural numbers and ℓ⩽kℓ⩽k such that mimi is odd for each i   with ℓ+1⩽i⩽kℓ+1⩽i⩽k. If G   is 45(m1+m2+⋯+mk)45(m1+m2+⋯+mk)-connected, then either G has a vertex set X   of order at most 2k+2ℓ−3+δ(m1,…,mℓ)2k+2ℓ−3+δ(m1,…,mℓ) such that G−XG−X is bipartite or G is modulo  (2m1,…,2mℓ,mℓ+1,…,mk)(2m1,…,2mℓ,mℓ+1,…,mk)-linked, whereδ(m1,…,mℓ)={0if min{m1,…,mℓ}=1,1if min{m1,…,mℓ}⩾2. Our results generalize several known results on parity-linked graphs.