Odd Edge Colorings of Graphs with Odd Order
Mikio Kano, Shun-ichi Maezawa, Kenta Ozeki

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 is said to be {\em odd -edge-colorable} if there exists an edge-coloring such that each non-empty color class induces an odd subgraph of . The {\em odd chromatic index} of , denoted by , is the minimum for which is odd -edge-colorable. In this paper, we prove that every -connected simple graph of odd order is odd 3-edge-colorable, and show that the -connectedness assumption is necessary. We also prove that for a connected Eulerian graph of odd order, there exists an edge such that is odd -edge-colorable.
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