General second-regime case of the Bougard–Joret formula

Establish the Bougard–Joret lower bound e(G) ≥ t(n, α) + ⌈kα/2⌉ for k-connected graphs of order n and independence number α in the general second regime n > kα, either by proving that the vertex set admits a partition into α nonempty cliques or by deriving the bound without assuming such a partition.

Background

The paper studies f(n, α, k), the minimum number of edges in a k-connected graph of order n and independence number α. Bougard and Joret conjectured a piecewise formula whose second line is t(n, α) + ⌈kα/2⌉ when n > kα. Prior work established this formula under certain large-order hypotheses, using a clique-partition theorem.

The paper disproves the conjectured first-line formula on the boundary n = α + k, but it does not resolve the second regime. Proposition 4.2 shows that the desired second-line bound follows whenever the vertices can be partitioned into α nonempty cliques. The unresolved issue is therefore to establish that bound in general, either by proving the requisite clique partition or by obtaining the same inequality through another argument.