A further extension of Yap's construction for ?-critical graphs

Yap introduced in [12] (J. Graph Theory (5 (1981) 159) a method for constructing -critical graphs. The construction has been extended by Plantholt [8] (J. Graph Theory 9 (1985) 371), Hilton and Zhao [5] (Discrete Appl. Math. 76 (1997) 205), and McMichael [6] (Preprint). It was shown in [5] that if G∗ is a graph obtained by splitting any vertex into two vertices from a -regular class 1 graph G, where ¿ 2 ( √ 7− 1)|G|+ 2, then G∗ is -critical. In this note, we further extend Yap’s construction by establishing the following result: if G∗ is a graph obtained from a -regular class 1 graph G, where ¿ |G|=2, by splitting any vertex of G into two vertices x and y such that x and y are minor vertices in G∗, then G∗ is -critical. c © 2002 Elsevier Science B.V. All rights reserved.