A further extension of Yapʹs construction for Δ-critical graphs

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⩾12(7−1)|G|+2, then G∗ is Δ-critical. In this note, we further extend Yapʹs 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.