Weak-odd chromatic index of special digraph classes

TL;DR

This paper introduces the concept of weak-odd chromatic index for digraphs, providing characterizations for semicomplete digraphs and extended tournaments, and explores weak-odd edge covering in digraphs.

Contribution

It generalizes existing results from tournaments to broader classes like semicomplete digraphs and extended tournaments, and initiates the study of weak-odd edge covering.

Findings

Characterization of weak-odd chromatic index for semicomplete digraphs

Characterization of weak-odd chromatic index for extended tournaments

Introduction of weak-odd edge covering in digraphs

Abstract

Give a digraph $D=(V(D),A(D))$, let $\partial^+_D(v)=\{vw|w\in N^+_D(v)\}$ and $\partial^-_D(v)=\{uv|u\in N^-_D(v)\}$ be semi-cuts of $v$. A mapping $\varphi:A(D)\rightarrow [k]$ is called a weak-odd $k$-edge coloring of $D$ if it satisfies the condition: for each $v\in V(D)$, there is at least one color with an odd number of occurrences on each non-empty semi-cut of $v$. We call the minimum integer $k$ the weak-odd chromatic index of $D$. When limit to 2 colors, use $def(D)$ to denote the defect of $D$, the minimum number of vertices in $D$ at which the above condition is not satisfied. In this paper, we give a descriptive characterization about the weak-odd chromatic index and the defect of semicomplete digraphs and extended tournaments, which generalize results of tournaments to broader classes. And we initiated the study of weak-odd edge covering on digraphs.