# On characterization of uniquely 3-list colorable complete multipartite graphs

Discussiones Mathematicae Graph Theory (2010)

- Volume: 30, Issue: 1, page 105-114
- ISSN: 2083-5892

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topYancai Zhao, and Erfang Shan. "On characterization of uniquely 3-list colorable complete multipartite graphs." Discussiones Mathematicae Graph Theory 30.1 (2010): 105-114. <http://eudml.org/doc/270997>.

@article{YancaiZhao2010,

abstract = {For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_\{2,2,r\}$ r ∈ 4,5,6,7,8, $K_\{2,3,4\}$, $K_\{1*4,4\}$, $K_\{1*4,5\}$, $K_\{1*5,4\}$. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_\{2,2,r\}$ r ∈ 4,5,6,7,8, the others have been proved not to be U3LC graphs. In this paper we first prove that $K_\{2,2,8\}$ is not U3LC graph, and thus as a direct corollary, $K_\{2,2,r\}$ (r = 4,5,6,7,8) are not U3LC graphs, and then the uniquely 3-list colorable complete multipartite graphs are characterized completely.},

author = {Yancai Zhao, Erfang Shan},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {list coloring; complete multipartite graph; uniquely 3-list colorable graph},

language = {eng},

number = {1},

pages = {105-114},

title = {On characterization of uniquely 3-list colorable complete multipartite graphs},

url = {http://eudml.org/doc/270997},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Yancai Zhao

AU - Erfang Shan

TI - On characterization of uniquely 3-list colorable complete multipartite graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2010

VL - 30

IS - 1

SP - 105

EP - 114

AB - For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2,2,r}$ r ∈ 4,5,6,7,8, $K_{2,3,4}$, $K_{1*4,4}$, $K_{1*4,5}$, $K_{1*5,4}$. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2,2,r}$ r ∈ 4,5,6,7,8, the others have been proved not to be U3LC graphs. In this paper we first prove that $K_{2,2,8}$ is not U3LC graph, and thus as a direct corollary, $K_{2,2,r}$ (r = 4,5,6,7,8) are not U3LC graphs, and then the uniquely 3-list colorable complete multipartite graphs are characterized completely.

LA - eng

KW - list coloring; complete multipartite graph; uniquely 3-list colorable graph

UR - http://eudml.org/doc/270997

ER -

## References

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