Geodesic
Unique list colorability of the graph $K^n_2+K_r$
Prikladnaâ diskretnaâ matematika, no. 1 (2022), pp. 88-94

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Given a list $L(v)$ for each vertex $v$, we say that the graph $G$ is $L$-colorable if there is a proper vertex coloring of $G$, where each vertex $v$ takes its color from $L(v)$. The graph is uniquely $k$-list colorable if there is a list assignment $L$ such that $|L(v)| = k$ for every vertex $v$ and the graph has exactly one $L$-coloring with these lists. If a graph $G$ is not uniquely $k$-list colorable, we also say that $G$ has property $M(k)$. The least integer $k$ such that $G$ has the property $M(k)$ is called the $m$-number of $G$, denoted by $m(G)$. In this paper, we characterize the unique list colorability of the graph $G=K^n_2+K_r$. In particular, we determine the number $m(G)$ of the graph $G=K^n_2+K_r$.
Keywords: vertex coloring, list coloring, uniquely list colorable graph, complete $r$-partite graph. 
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     author = {L. X. Hung},
     title = {Unique list colorability of the graph $K^n_2+K_r$},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {88--94},
     publisher = {mathdoc},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PDM_2022_1_a5/}
}
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L. X. Hung. Unique list colorability of the graph $K^n_2+K_r$. Prikladnaâ diskretnaâ matematika, no. 1 (2022), pp. 88-94. http://geodesic.mathdoc.fr/item/PDM_2022_1_a5/