Every 2-connected graph has a cycle through all vertices of degree at least its bipartite-hole-number, and high-degree pairs are joined by a path through all such vertices.
Dirac, Some theorems on abstract graphs,Proc
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Cycles and paths through vertices whose degrees are at least the bipartite-hole-number
Every 2-connected graph has a cycle through all vertices of degree at least its bipartite-hole-number, and high-degree pairs are joined by a path through all such vertices.