On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs
A graph $G$ is called uniquely k-list colorable (U$k$LC) if there exists a list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $, each of size $k$, such that there is a unique proper list coloring of $G$ from this list of colors. A graph $G$ is said to have property $M(k)$ if...
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| creator | Abdolmaleki, M Hutchinson, J. P Ilchi, S. Gh Mahmoodian, E. S Shabani, M. A |
| description | A graph $G$ is called uniquely k-list colorable (U$k$LC) if there exists a
list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $,
each of size $k$, such that there is a unique proper list coloring of $G$ from
this list of colors. A graph $G$ is said to have property $M(k)$ if it is not
uniquely $k$-list colorable. Mahmoodian and Mahdian characterized all graphs
with property $M(2)$. For $k\geq 3$ property $M(k)$ has been studied only for
multipartite graphs. Here we find bounds on $M(k)$ for graphs embedded on
surfaces, and obtain new results on planar graphs. We begin a general study of
bounds on $M(k)$ for regular graphs, as well as for graphs with varying list
sizes. |
| doi_str_mv | 10.48550/arxiv.1705.07434 |
| format | Article |
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list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $,
each of size $k$, such that there is a unique proper list coloring of $G$ from
this list of colors. A graph $G$ is said to have property $M(k)$ if it is not
uniquely $k$-list colorable. Mahmoodian and Mahdian characterized all graphs
with property $M(2)$. For $k\geq 3$ property $M(k)$ has been studied only for
multipartite graphs. Here we find bounds on $M(k)$ for graphs embedded on
surfaces, and obtain new results on planar graphs. We begin a general study of
bounds on $M(k)$ for regular graphs, as well as for graphs with varying list
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list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $,
each of size $k$, such that there is a unique proper list coloring of $G$ from
this list of colors. A graph $G$ is said to have property $M(k)$ if it is not
uniquely $k$-list colorable. Mahmoodian and Mahdian characterized all graphs
with property $M(2)$. For $k\geq 3$ property $M(k)$ has been studied only for
multipartite graphs. Here we find bounds on $M(k)$ for graphs embedded on
surfaces, and obtain new results on planar graphs. We begin a general study of
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list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $,
each of size $k$, such that there is a unique proper list coloring of $G$ from
this list of colors. A graph $G$ is said to have property $M(k)$ if it is not
uniquely $k$-list colorable. Mahmoodian and Mahdian characterized all graphs
with property $M(2)$. For $k\geq 3$ property $M(k)$ has been studied only for
multipartite graphs. Here we find bounds on $M(k)$ for graphs embedded on
surfaces, and obtain new results on planar graphs. We begin a general study of
bounds on $M(k)$ for regular graphs, as well as for graphs with varying list
sizes.</abstract><doi>10.48550/arxiv.1705.07434</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics |
| title | On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs |
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