A sufficient condition for a graph to be fractional (k,n)-critical

Abstract

Toughness implies the topological representation characterization of network graphs. The pioneering works implicated that there are salient relationship between toughness and the existence of fractional factor in various settings. Specifically, Liu and Zhang (2008) [7] determined that G admits a fractional k-factor if t(G)≥k−1k and |V(G)|≥k+1 (k≥2 is an integer), and the toughness bound is tight. A graph G is fractional (k,n)-critical, if removing any n vertices from G the resulting subgraph still admits a fractional k-factor. In this paper, we prove that if k>n, then the original toughness condition (i.e., t(G)≥k−1k) is sufficient for a graph to be fractional (k,n)-critical. Furthermore, we explain why k>n is the extreme boundary to keep the original toughness bound.