A characterization on (g,f)-parity orientations

Abstract

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