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New isolated toughness condition for fractional $(g,f,n)$-critical graphs

Volume 147 / 2017

Wei Gao, Weifan Wang Colloquium Mathematicum 147 (2017), 55-65 MSC: Primary 05C70. DOI: 10.4064/cm6713-8-2016 Published online: 8 December 2016


Let $i(G)$ be the number of isolated vertices in a graph $G$. The isolated toughness of $G$ is defined as $I(G)=\infty $ if $G$ is complete, and $I(G)=\operatorname{min}\{|S|/i(G-S) : S\subseteq V(G),\, i(G-S)\ge 2\}$ otherwise. We show that $G$ is a fractional $(g,f,n)$-critical graph if $I(G)\ge (b^{2}+bn-\varDelta )/{a}$, where $a, b$ are positive integers, $1\le a\le b$, $b\ge 2$, and $\varDelta =b-a$. Furthermore, a new isolated toughness condition for fractional $(a,b,n)$-critical graphs is given.


  • Wei GaoSchool of Information Science and Technology
    Yunnan Normal University
    Kunming 650500, China
  • Weifan WangDepartment of Mathematics
    Zhejiang Normal University
    Jinhua 321004, China

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