Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916109485080576 |
|---|---|
| author | Anto, Nevil Basavaraju, Manu Hegde, Suresh Manjanath Kulamarva, Shashanka |
| author_facet | Anto, Nevil Basavaraju, Manu Hegde, Suresh Manjanath Kulamarva, Shashanka |
| contents | An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamč\'ık conjectured that $a'(G) \le Δ+2$ for any graph $G$ with maximum degree $Δ$. A graph $G$ is said to be $k$-degenerate if every subgraph of $G$ has a vertex of degree at most $k$. Basavaraju and Chandran proved that the conjecture is true for $2$-degenerate graphs. We prove that for a $3$-degenerate graph $G$, $a'(G) \le Δ+5$, thereby bringing the upper bound closer to the conjectured bound. We also consider $k$-degenerate graphs with $k \ge 4$ and give an upper bound for the acyclic chromatic index of the same. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01948 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs Anto, Nevil Basavaraju, Manu Hegde, Suresh Manjanath Kulamarva, Shashanka Combinatorics Discrete Mathematics An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamč\'ık conjectured that $a'(G) \le Δ+2$ for any graph $G$ with maximum degree $Δ$. A graph $G$ is said to be $k$-degenerate if every subgraph of $G$ has a vertex of degree at most $k$. Basavaraju and Chandran proved that the conjecture is true for $2$-degenerate graphs. We prove that for a $3$-degenerate graph $G$, $a'(G) \le Δ+5$, thereby bringing the upper bound closer to the conjectured bound. We also consider $k$-degenerate graphs with $k \ge 4$ and give an upper bound for the acyclic chromatic index of the same. |
| title | Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2305.01948 |