Conflict-free chromatic index of trees
A graph $G$ is conflict-free $k$-edge-colorable if there exists an assignment of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color that is assigned to exactly one edge among the closed neighborhood of $e$. The smallest $k$ such that $G$ is conflict-free $k$-edge-colorable i...
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| creator | Guo, Shanshan Li, Ethan Y. H Li, Luyi Li, Ping |
| description | A graph $G$ is conflict-free $k$-edge-colorable if there exists an assignment
of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color
that is assigned to exactly one edge among the closed neighborhood of $e$. The
smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the
conflict-free chromatic index of $G$, denoted $\chi'_{CF}(G)$. D\c{e}bski and
Przyby\a{l}o showed that $2\le\chi'_{CF}(T)\le 3$ for every tree $T$ of size at
least two. In this paper, we present an algorithm to determine the
conflict-free chromatic index of a tree without 2-degree vertices, in time
$O(|V(T)|)$. This partially answer a question raised by Kamyczura, Meszka and
Przyby\a{l}o. |
| doi_str_mv | 10.48550/arxiv.2409.10899 |
| format | Article |
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of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color
that is assigned to exactly one edge among the closed neighborhood of $e$. The
smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the
conflict-free chromatic index of $G$, denoted $\chi'_{CF}(G)$. D\c{e}bski and
Przyby\a{l}o showed that $2\le\chi'_{CF}(T)\le 3$ for every tree $T$ of size at
least two. In this paper, we present an algorithm to determine the
conflict-free chromatic index of a tree without 2-degree vertices, in time
$O(|V(T)|)$. This partially answer a question raised by Kamyczura, Meszka and
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of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color
that is assigned to exactly one edge among the closed neighborhood of $e$. The
smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the
conflict-free chromatic index of $G$, denoted $\chi'_{CF}(G)$. D\c{e}bski and
Przyby\a{l}o showed that $2\le\chi'_{CF}(T)\le 3$ for every tree $T$ of size at
least two. In this paper, we present an algorithm to determine the
conflict-free chromatic index of a tree without 2-degree vertices, in time
$O(|V(T)|)$. This partially answer a question raised by Kamyczura, Meszka and
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of $k$ colors to $E(G)$ such that for every edge $e\in E(G)$, there is a color
that is assigned to exactly one edge among the closed neighborhood of $e$. The
smallest $k$ such that $G$ is conflict-free $k$-edge-colorable is called the
conflict-free chromatic index of $G$, denoted $\chi'_{CF}(G)$. D\c{e}bski and
Przyby\a{l}o showed that $2\le\chi'_{CF}(T)\le 3$ for every tree $T$ of size at
least two. In this paper, we present an algorithm to determine the
conflict-free chromatic index of a tree without 2-degree vertices, in time
$O(|V(T)|)$. This partially answer a question raised by Kamyczura, Meszka and
Przyby\a{l}o.</abstract><doi>10.48550/arxiv.2409.10899</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Computer Science - Discrete Mathematics Mathematics - Combinatorics |
| title | Conflict-free chromatic index of trees |
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