Ore-type degree conditions, Path-Systems and Linkages in Graphs January 14, 2003 Ronald J. Gould, Thor C. Whalen (Emory University) Abstract Let G be a graph of order n and σ(G) = min{d(u)+ d(v): uv / ∈ E(G)}. In this paper we prove that if σ(G) ≥ n + k − 2 then G is Hamiltonian k-connected and if σ(G) ≥ n +2k − 3 then is Hamiltonian k-linked. We prove similar results for balanced bipartite graphs. All these results are tight. 1 Preliminaries Let G =(V (G),E(G)) be a simple loopless graph. For two given subgraphs F and H of G, we denote by F H the subgraph of G induced by the vertices of F in H (i.e. the graph with vertex set V (F ) and edge set E(F ) ∩ E(H ). For any given subgraphs A and B of G, let N (A) be the set of vertices of G that are adjacent to at least one vertex of A, let E(A, B) be the set of edges that have an end vertex in A and the other in B, let d(A, B)= |E(A, B)| (so for a given vertex v ∈ V (A), d(v,B)= N (v) ∩ V (B), let δ (A)= min v∈V (A) {d(v)} let δ (A, B)= min v∈V (A) {d(v,B)}. Let σ ( G) = min{d(x)+ d(y): xy / ∈ E(G)}. Let P =(z 1 , ··· ,z m ) be a path of G and z = z i be a vertex of V (P ). If i ≥ 2, the predecessor z − of z is the vertex z i−1 . If i ≤ n − 1, the successor z + of z is the vertex z i+1 . If A is a subset of V (P ), then A − (resp. A + ) will denote the set of of predecessors (resp. successors) of the vertices of A. 1