A characterization on $(g,f)$-parity orientations

Abstract

Let $G$ be a graph and $g,f:V(G)\to 2^N$ be two set functions such that $g(v)\le f(v)$ and $g(v)\equiv f(v)(\mathrm{mod}2)$ for every $v\in V(G)$. An orientation $O$ of $G$ is called a $(g,f)$-parity orientation if $g(v)\le d_O^{+}(v)\le f(v)$ and $g(v)\equiv d_O^{+}(v)(\mathrm{mod}2)$ for every $v\in V(G)$. In this paper, we give a Tutte-type characterization for a graph to have a $(g,f)$-parity orientation.