The Tutte's condition in terms of graph factors

TL;DR

This paper characterizes when connected graphs satisfy Tutte's condition based on a function and relates it to the existence of specific graph factors, extending classical results to more general settings.

Contribution

It provides a new characterization of graphs satisfying Tutte's condition in terms of the existence of certain $H$-factors, generalizing previous conditions.

Findings

Characterization of graphs satisfying Tutte's condition via $H$-factors

Extension of Tutte's condition to graphs of odd order

Conditions involving functions $J_f(v)$ and $J_f^+(v)$

Abstract

Let $G$ be a connected general graph of even order, with a function $f\colon V(G)\to\Z^+$. 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 $H\colon V(G)\to 2^\N$ 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.