On Tuckerʹs proof of the strong perfect graph conjecture for (K4−e)-free graphs

In this note, the authors generalize the ideas presented by Tucker in his proof of the Strong Perfect Graph Conjecture for (K4−e)-free graphs in order to find a vertex v in G whose special neighborhood allows to extend a ω(G)-vertex coloring of G−v to a ω(G)-vertex coloring of G. The search for such a vertex led us to the definition of p-Tucker vertices: vertices contained in at most two maximal cliques of size at least p; and to a family of classes Gp of graphs G whose maximal cliques of size at least p have no edge in common with any other maximal cliques of G. We prove that every hole-free graph G in Gp has a p-Tucker vertex and we use this fact to compute ω(G) in polynomial time for each class Gp. We state a conjecture whose validity yields the validity of the Strong Perfect Graph Conjecture for (K5−e)-free graphs clique.