Odd edge-colorings of subdivisions of odd graphs

TL;DR

This paper characterizes subdivisions of odd graphs in terms of their odd chromatic index, extending previous results and supporting the conjecture that all such graphs are odd 4-edge-colorable with a small color class.

Contribution

It extends the characterization of subcubic graphs to all subdivisions of odd graphs based on their odd chromatic index, and proves the conjecture for this class.

Findings

Every connected subdivision of an odd graph with maximum four colors becomes odd 3-edge-colorable after removing one edge.

Supports the conjecture that all subdivisions of odd graphs are odd 4-edge-colorable with a color class of size at most 1.

Provides a characterization of all subdivisions of odd graphs in terms of the odd chromatic index.

Abstract

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum value of $k$ for which such a decomposition of $G$ exists is the odd chromatic index, $\chi_{o}'(G)$, introduced by Pyber (1991). For every $k\geq\chi_{o}'(G)$, the graph $G$ is said to be odd $k$-edge-colorable. Apart from two particular exceptions, which are respectively odd $5$- and odd $6$-edge-colorable, the rest of connected loopless graphs are odd $4$-edge-colorable, and moreover one of the color classes can be reduced to size $\leq2$. In addition, it has been conjectured that an odd $4$-edge-coloring with a color class of size at most $1$ is always achievable. Atanasov et al. (2016) characterized the class of subcubic graphs in terms of the value $\chi_{o}'(G)\leq4$. In this paper, we extend their result to a characterization of all subdivisions of odd graphs in terms of the value of the odd chromatic index. This larger class $\mathcal{S}$ is of a particular interest as it collects all `least instances' of non-odd graphs. As a prelude to our main result, we show that every connected graph $G\in \mathcal{S}$ requiring the maximum number of four colors, becomes odd $3$-edge-colorable after removing a certain edge. Thus, we provide support for the mentioned conjecture by proving it for all subdivisions of odd graphs. The paper concludes with few problems for possible further work.