Acyclic Edge Coloring of 3-sparse Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911554533851136 |
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| author | Anto, Nevil Basavaraju, Manu Kulamarva, Shashanka |
| author_facet | Anto, Nevil Basavaraju, Manu Kulamarva, Shashanka |
| contents | A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamč\'ık conjectured that for a graph $G$ with maximum degree $Δ$, $a'(G) \le Δ+2$. A graph $G$ is said to be $3$-sparse if every edge in $G$ is incident on at least one vertex of degree at most $3$. We prove the conjecture for the class of $3$-sparse graphs. Further, we give a stronger bound of $Δ+1$, if there exists an edge $xy$ in the graph with $d_G(x)+ d_G(y) < Δ+3$. When $ Δ> 3$, the $3$-sparse graphs where no such edge exists is the set of bipartite graphs where one partition has vertices with degree exactly $3$ and the other partition has vertices with degree exactly $Δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Acyclic Edge Coloring of 3-sparse Graphs Anto, Nevil Basavaraju, Manu Kulamarva, Shashanka Combinatorics Discrete Mathematics 05C15 A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamč\'ık conjectured that for a graph $G$ with maximum degree $Δ$, $a'(G) \le Δ+2$. A graph $G$ is said to be $3$-sparse if every edge in $G$ is incident on at least one vertex of degree at most $3$. We prove the conjecture for the class of $3$-sparse graphs. Further, we give a stronger bound of $Δ+1$, if there exists an edge $xy$ in the graph with $d_G(x)+ d_G(y) < Δ+3$. When $ Δ> 3$, the $3$-sparse graphs where no such edge exists is the set of bipartite graphs where one partition has vertices with degree exactly $3$ and the other partition has vertices with degree exactly $Δ$. |
| title | Acyclic Edge Coloring of 3-sparse Graphs |
| topic | Combinatorics Discrete Mathematics 05C15 |
| url | https://arxiv.org/abs/2501.11281 |