Odd Edge Colorings of Graphs with Odd Order

TL;DR

This paper investigates odd edge colorings in graphs, proving that 4-connected graphs of odd order are odd 3-edge-colorable and that Eulerian graphs of odd order can be made odd 2-edge-colorable by removing one edge.

Contribution

It establishes new bounds on odd edge colorings for 4-connected graphs and Eulerian graphs of odd order, highlighting the necessity of connectivity assumptions.

Findings

Every 4-connected simple graph of odd order is odd 3-edge-colorable.

The 4-connectedness condition is necessary for the main result.

In Eulerian graphs of odd order, removing one edge yields an odd 2-edge-colorable graph.

Abstract

An {\em odd subgraph} of a graph is a subgraph in which every vertex has odd degree. A graph $G$ is said to be {\em odd $k$-edge-colorable} if there exists an edge-coloring $E(G) \rightarrow \{1,2, \ldots, k\}$ such that each non-empty color class induces an odd subgraph of $G$. The {\em odd chromatic index} of $G$, denoted by $\chi'_o(G)$, is the minimum $k$ for which $G$ is odd $k$-edge-colorable. In this paper, we prove that every $4$-connected simple graph of odd order is odd 3-edge-colorable, and show that the $4$-connectedness assumption is necessary. We also prove that for a connected Eulerian graph $G$ of odd order, there exists an edge $e$ such that $G-e$ is odd $2$-edge-colorable.