On vertex-parity edge-colorings

Abstract

A vertex signature $\pi$ of a finite graph G is any mapping $\pi \,{:}\,V(G)\rightarrow \{0,1\}$. An edge-coloring of G is said to be vertex-parity for the pair $(G,\pi )$ if for every vertex v each color used on the edges incident to v appears in parity accordance with $\pi$, i.e. an even or odd number of times depending on whether $\pi (v)$ equals 0 or 1, respectively. The minimum number of colors for which $(G,\pi )$ admits such an edge-coloring is denoted by $\chi '_p(G,\pi )$. We characterize the existence and prove that $\chi '_p(G,\pi )$ is at most 6. Furthermore, we give a structural characterization of the pairs $(G,\pi )$ for which $\chi '_p(G,\pi )=5$ and $\chi '_p(G,\pi )=6$. In the last part of the paper, we consider a weaker version of the coloring, where it suffices that at every vertex, at least one color appears in parity accordance with $\pi$. We show that the corresponding chromatic index is at most 3 and give a complete characterization for it.