Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.10192 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915357812326400 |
|---|---|
| author | Liu, Xiao-Chuan Petruševski, Mirko Yang, Xu |
| author_facet | Liu, Xiao-Chuan Petruševski, Mirko Yang, Xu |
| contents | An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per edge is allowed, we speak of an odd $k$-edge-covering of $G$. In this paper, we fully resolve two major conjectures on odd edge-colorings and odd edge-coverings of graphs, proposed by Petru{š}evski and {Š}krekovski ({\it European Journal of Combinatorics,} 91:103225, 2021). The first conjecture states that, apart from two particular exceptions which are respectively odd $5$- and odd-$6$-edge-colorable, for any other loopless and connected graph $G$ there exists an edge $e$ such that $G\backslash \{e\}$ is odd $3$-edge-colorable. The second conjecture states that any simple graph $G$ admits an odd $3$-edge-covering in which at most one edge receives more than one color. In addition, we strongly confirm the second conjecture by demonstrating that there exists an odd $3$-edge-covering in which at most one edge receives two colors and the rest of the edges receive unique colors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Graph Odd Edge-Colorings and Odd Edge-Coverings Liu, Xiao-Chuan Petruševski, Mirko Yang, Xu Combinatorics An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per edge is allowed, we speak of an odd $k$-edge-covering of $G$. In this paper, we fully resolve two major conjectures on odd edge-colorings and odd edge-coverings of graphs, proposed by Petru{š}evski and {Š}krekovski ({\it European Journal of Combinatorics,} 91:103225, 2021). The first conjecture states that, apart from two particular exceptions which are respectively odd $5$- and odd-$6$-edge-colorable, for any other loopless and connected graph $G$ there exists an edge $e$ such that $G\backslash \{e\}$ is odd $3$-edge-colorable. The second conjecture states that any simple graph $G$ admits an odd $3$-edge-covering in which at most one edge receives more than one color. In addition, we strongly confirm the second conjecture by demonstrating that there exists an odd $3$-edge-covering in which at most one edge receives two colors and the rest of the edges receive unique colors. |
| title | On Graph Odd Edge-Colorings and Odd Edge-Coverings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.10192 |