Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Journal of Graph Theory
On k-ordered Hamiltonian graphs
Journal of Graph Theory
Journal of Graph Theory
Degree conditions for k-ordered hamiltonian graphs
Journal of Graph Theory
An improved linear edge bound for graph linkages
European Journal of Combinatorics - Special issue: Topological graph theory II
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
On the connectivity of minimum and minimal counterexamples to Hadwiger's Conjecture
Journal of Combinatorial Theory Series B
Non-zero disjoint cycles in highly connected group labelled graphs
Journal of Combinatorial Theory Series B
Minimum degree conditions for H-linked graphs
Discrete Applied Mathematics
Linkedness and ordered cycles in digraphs
Combinatorics, Probability and Computing
Linked graphs with restricted lengths
Journal of Combinatorial Theory Series B
The extremal function for 3-linked graphs
Journal of Combinatorial Theory Series B
k-Ordered Hamilton cycles in digraphs
Journal of Combinatorial Theory Series B
Linear connectivity forces large complete bipartite minors
Journal of Combinatorial Theory Series B
Linear connectivity forces large complete bipartite minors
Journal of Combinatorial Theory Series B
FAW '09 Proceedings of the 3d International Workshop on Frontiers in Algorithmics
European Journal of Combinatorics
A Fan-type result on k-ordered graphs
Information Processing Letters
Algorithms for finding a maximum non-k-linked graph
ESA'11 Proceedings of the 19th European conference on Algorithms
Linkage for the diamond and the path with four vertices
Journal of Graph Theory
New Ore-Type Conditions for H-Linked Graphs
Journal of Graph Theory
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A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.