The Tutte's condition in terms of graph factors

Abstract

Let $G$ be a connected general graph of even order, with a function $f:V(G)\to {\mathbb{Z}}^{+}$. We obtain that $G$ satisfies the Tutte's condition \[ o(G-S)\le \sum_{v\in S}f(v)\qquad\text{for any nonempty set $S\subset V(G)$}, \] with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H:V(G)\to 2^{\mathbb{N}}$ such that $H(v)\in \{J_f(v),J_f^{+}(v)\}$ for each $v\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J_f^{+}(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a characterization for graphs of odd order satisfying the Tutte's condition with respect to a function.

5 pages