A Tutte-type characterization for graph factors

TL;DR

This paper generalizes Tutte-type conditions for graph factors, providing new characterizations involving colored factors and addressing open problems in graph theory related to odd factors and conditions.

Contribution

It introduces a Tutte-type characterization for the existence of colored $J_f^*$-factors in graphs, extending previous theorems and solving open problems.

Findings

Characterizes graphs satisfying $o(G-S) \\le f(S)$ with colored $J_f^*$-factors

Provides an analogous characterization for graphs of odd order

Addresses open problems by Cui, Kano, Akiyama

Abstract

Let $G$ be a connected general graph. Let $f\colon V(G)\to \Z^+$ be a function. We show that $G$ satisfies the Tutte-type condition \[ o(G-S)\le f(S)\qquad\text{for all vertex subsets $S$}, \] if and only if it contains a colored $J_f^*$-factor for any $2$-end-coloring, where $J_f^*(v)$ is the union of all odd integers smaller than $f(v)$ and the integer $f(v)$ itself. This is a generalization of the $(1,f)$-odd factor characterization theorem, and answers a problem of Cui and Kano. We also derive an analogous characterization for graphs of odd orders, which addresses a problem of Akiyama and Kano.