On the graphs with distinguishing number equal list distinguishing number

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Alikhani Saeid; Soltani Samaneh

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Journal Paper

Paper Information

Journal: Journal of Mahani Mathematical Research Year:2023 | Volume:12 | Issue:2 Page(s): 411-423

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Information Journal Paper

Title

On the graphs with distinguishing number equal list distinguishing number

Author(s)

Alikhani Saeid | Soltani Samaneh | Issue Writer Certificate

Keywords

Distinguishing numberQ4

list-distinguishing labelingQ4

list distinguishing chromatic numberQ4

Abstract

The Distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by the trivial automorphism. A list assignment to $G$ is an assignment $L = \{L(v)\}_{v\in V (G)}$ of lists of labels to the vertices of $G$. A distinguishing $L$-labeling of $G$ is a distinguishing labeling of $G$ where the label of each vertex $v$ comes from $L(v)$. The list Distinguishing number of $G$, $D_l(G)$ is the minimum $k$ such that every list assignment to $G$ in which $|L(v)| = k$ for all $v \in V (G)$ yields a distinguishing $L$-labeling of $G$. In this paper, we determine the list-Distinguishing number for two families of graphs. We also characterize graphs with the Distinguishing number equal the list Distinguishing number. Finally, we show that this characterization works for other list numbers of a graph.

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