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

Abstract Yap introduced in [12] (J. Graph Theory (5 (1981) 159) a method for constructing Δ-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 Hilton [5] that if G ∗ is a graph obtained by splitting any vertex into two vertices from a Δ-regular class 1 graph G, where Δ⩾ 1 2 ( 7 −1)|G|+2 , then G ∗ is Δ-critical. In this note, we further extend Yap's construction by establishing the following result: if G ∗ is a graph obtained from a Δ-regular class 1 graph G, where Δ⩾|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 Δ-critical.