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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{PDM_2022_1_a5,
     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/}
}
TY  - JOUR
AU  - L. X. Hung
TI  - Unique list colorability of the graph $K^n_2+K_r$
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2022
SP  - 88
EP  - 94
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2022_1_a5/
LA  - en
ID  - PDM_2022_1_a5
ER  - 
%0 Journal Article
%A L. X. Hung
%T Unique list colorability of the graph $K^n_2+K_r$
%J Prikladnaâ diskretnaâ matematika
%D 2022
%P 88-94
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2022_1_a5/
%G en
%F PDM_2022_1_a5
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/

[1] M. Behzad, G. Chartrand, Introduction to the Theory of Graphs, Allyn and Bacon, Boston, 1971 | MR | Zbl

[2] V. G. Vizing, “Coloring the vertices of a graph in prescribed colors”, Diskret. Analiz, 29 (1976), 3–10 (in Russian) | MR | Zbl

[3] P. Erdös, A. L. Rubin, H. Taylor, “Choosability in graphs”, Proc. West Coast Conf. Combinatorics, Graph Theory, Computing (Arcata, CA, September 1979), Congr. Numer., 26, 125–157 | MR

[4] J. H. Dinitz, W. J. Martin, “The stipulation polynomial of a uniquely list colorable graph”, Australasian J. Combin., 11 (1995), 105–115 | MR | Zbl

[5] M. Mahdian, E. S. Mahmoodian, “A characterization of uniquely 2-list colorable graphs”, Ars Combin., 51 (1999), 295–305 | MR | Zbl

[6] W. Wang, X. Liu, “List-coloring based channel allocation for open-spectrum wireless networks”, Proc. VTC'05, 2005, 690–694

[7] L. X. Hung, “List-chromatic number and chromatically unique of the graph $K^r_2+O_k$”, Selecciones Matematicas, Universidad Nacional de Trujillo, 06:01 (2019), 26–30

[8] L. X. Hung, “Colorings of the graph $K^m_2 + K_n$”, J. Sib. Fed. Univ. Math. Phys., 13:3 (2020), 297–305 | DOI | MR | Zbl

[9] M. Ghebleh, E. S. Mahmoodian, “On uniquely list colorable graphs”, Ars Combin., 59 (2001), 307–318 | MR | Zbl