Characterizing 2k-critical graphs and n-extendable graphs

Let G be a graph with even order. Let M be a matching in G and x1, x2, . . . , x2r , be the M-unsaturated vertices in G. Then G has a perfect matching if and only if there are r independentM-augmenting paths joining the 2r vertices in pairs. LetG be a graph with a perfect matchingM . It is proved that G is 2k-critical if and only if for any 2k vertices u1, u2, . . . , u2k in G, there are k independentM-alternating paths P1, P2, . . . , Pk joining the 2k vertices in pairs such that P1, P2, . . . , Pk start and end with edges inM . It is also proved thatG is n-extendable if and only if, for each r with 0 r n and each F ⊂ M with |F |= r , and for any n− r pairs ofM-alternating paths xix′ iy′ iyi with xix′ i , yiy′ i ∈ M (1 i n− r) in G− V (F), there exist independentMalternating paths P1, P2, . . . , Pk inG− V (F) joining the vertices in Z = {x1, y1, . . . , xn−r , yn−r }\{x′ 1, y′ 1, . . . , x′ n−r , y′ n−r }, where |Z| = 2k, which start and end with edges in E(G)\M . © 2004 Published by Elsevier B.V.