Linked graphs with restricted lengths

A graph G is k-linked if G has at least 2k vertices, and for every sequence x 1 , x 2 , … , x k , y 1 , y 2 , … , y k of distinct vertices, G contains k vertex-disjoint paths P 1 , P 2 , … , P k such that P i joins x i and y i for i = 1 , 2 , … , k . Moreover, the above defined k-linked graph G is modulo ( m 1 , m 2 , … , m k ) -linked if, in addition, for any k-tuple ( d 1 , d 2 , … , d k ) of natural numbers, the paths P 1 , P 2 , … , P k can be chosen such that P i has length d i modulo m i for i = 1 , 2 , … , k . Thomassen showed that, for each k-tuple ( m 1 , m 2 , … , m k ) of odd positive integers, there exists a natural number f ( m 1 , m 2 , … , m k ) such that every f ( m 1 , m 2 , … , m k ) -connected graph is modulo ( m 1 , m 2 , … , m k ) -linked. For m 1 = m 2 = … = m k = 2 , he showed in another article that there exists a natural number g ( 2 , k ) such that every g ( 2 , k ) -connected graph G is modulo ( 2 , 2 , … , 2 ) -linked or there is X ⊆ V ( G ) such that | X | ⩽ 4 k − 3 and G − X is a bipartite graph, where ( 2 , 2 , … , 2 ) is a k-tuple. wed that f ( m 1 , m 2 , … , m k ) ⩽ max { 14 ( m 1 + m 2 + ⋯ + m k ) − 4 k , 6 ( m 1 + m 2 + ⋯ + m k ) − 4 k + 36 } for every k-tuple of odd positive integers. We then extend the result to allow some m i be even integers. Let ( m 1 , m 2 , … , m k ) be a k-tuple of natural numbers and ℓ ⩽ k such that m i is odd for each i with ℓ + 1 ⩽ i ⩽ k . If G is 45 ( m 1 + m 2 + ⋯ + m k ) -connected, then either G has a vertex set X of order at most 2 k + 2 ℓ − 3 + δ ( m 1 , … , m ℓ ) such that G − X is bipartite or G is modulo ( 2 m 1 , … , 2 m ℓ , m ℓ + 1 , … , m k ) -linked, where δ ( m 1 , … , m ℓ ) = { 0 if min { m 1 , … , m ℓ } = 1 , 1 if min { m 1 , … , m ℓ } ⩾ 2 . Our results generalize several known results on parity-linked graphs.