A Note on Fragments in a Locally $k$-Critical $n$-Connected Graph

Abstract

We note that with only a slight modification, Su’s proof on the fragments in $k$-critical $n$-connected graphs (see J. Graph Theory $45(2004),281-297$) can imply the following more general result: every non-complete $W$-locally $k$-critical $n$-connected graph has $2k+2$ distinct fragments $F_1,F_2,\dots ,F_{2k+2}$ such that $F_1\cap W,F_2\cap W,\dots ,F_{2k+2}\cap W$ are pairwise disjoint.