An upper bound on the adaptable choosability of graphs.
International audience ; Given a (possibly improper) edge-colouring $F$ of a graph $G$, a vertex colouring $c$ of $G$ is \emph{adapted to} $F$ if no colour appears at the same time on an edge and on its two endpoints. If for some integer $k$, a graph...
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International audience ; Given a (possibly improper) edge-colouring $F$ of a graph $G$, a vertex colouring $c$ of $G$ is \emph{adapted to} $F$ if no colour appears at the same time on an edge and on its two endpoints. If for some integer $k$, a graph $G$ is such that given any list assignment $L$ of $G$, with $|L(v)| \ge k$ for all $v$, and any edge-colouring $F$ of $G$, there exists a vertex colouring $c$ of $G$ adapted to $F$ such that $c(v) \in L(v)$ for all $v$, then $G$ is said to be \emph{adaptably $k$-choosable}. The smallest $k$ such that $G$ is adaptably $k$-choosable is called the \emph{adaptable choice number} and is denoted by $ch_{ad}(G)$. This note proves that $ch_{ad}(G) \le \lceil {Mad}(G)/2 \rceil +1$, where ${Mad}(G)$ is the maximum of $2|E(H)|/|V(H)|$ over all subgraphs $H$ of $G$. As a consequence, we give bounds for classes of graphs embeddable into surfaces of non-negative Euler characteristics.
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Optimal area algorithm for planar polyline drawings.
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Orthogonal Drawings based on the Stratification of Planar Graphs
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