The core-forcing principle for perfect flowers

Sterboul's theorem characterizes non-K®nigEgerváry graphs by the presence, relative to a maximum matching, of a ower or a posy. In this paper we translate that obstruction into the language of perfect owers and the core of the graph. We introduce core-defective perfect owers: perfect owers whose alternating path contains a vertex at odd distance from the blossom base that does not belong to the core. We prove rst that every K®nig Egerváry graph is core-rigid: in every perfect ower, all odd-distance vertices of the attaching path lie in the core. Conversely, ifGis connected and is not an odd cycle, thenGis non-K®nigEgerváry if and only ifGcontains a core-defective perfect ower. Thus, among connected graphs dierent from an odd cycle, the K®nigEgerváry graphs are exactly the graphs with no core-defective perfect ower. In the matchable case the statement strengthens: ifGhas a perfect matching, then being non-K®nigEgerváry is equivalent to the existence of a core-defective perfect ower for some maximum matching, and also equivalent to the existence of one for every maximum matching. We include examples and counterexamples showing why odd cycles, disconnected graphs, and the universal quanti er over maximum matchings require separate treatment.