An improved linear edge bound for graph linkages
European Journal of Combinatorics - Special issue: Topological graph theory II
On Sufficient Degree Conditions for a Graph to be $k$-linked
Combinatorics, Probability and Computing
K-linked graphs with girth condition
Journal of Graph Theory
On the odd-minor variant of Hadwiger's conjecture
Journal of Combinatorial Theory Series B
Combinatorica
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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 pairwise vertex-disjoint paths P 1 ,P 2 ,...,P k such that P i joins x i and y i for i = 1,2,...,k . We say that G is modulo (m 1 ,m 2 ,...,m k )-linked if G is k -linked and, 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 [15] showed that if each m i is odd and G has sufficiently high connectivity then G is modulo (m 1 ,m 2 ,...,m k )-linked . In this paper, we will prove that when G is $(92\sum_{i=1}^k m_i-44k)$-connected, where m i is a prime or m i = 1, either G is modulo (2m 1 ,2m 2 ,..., 2m k )-linked or G has a vertex set X of order at most 4k *** 3 such that G *** X is bipartite.