Linear and 2-frugal choosability of graphs of small maximum average degree

Nathann Cohen (1) , Frédéric Havet (1)

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications

A proper vertex colouring of a graph $G$ is {\it 2-frugal} (resp. {\it linear}) if the graph induced by the vertices of any two colour classes is of maximum degree 2 (resp. is a forest of paths). A graph $G$ is {\it 2-frugally} (resp. {\it linearly}) {\it $L$-colourable} if for a given list assignment $L:V(G)\mapsto 2^{\mathbb N}$, there exists a 2-frugal (resp. linear) colouring $c$ of $G$ such that $c(v)\in L(v)$ for all $v\in V(G)$. If $G$ is 2-frugally (resp. linearly) $L$-list colourable for any list assignment such that $|L(v)|\ge k$ for all $v\inV(G)$, then $G$ is {\it 2-frugally} (resp. {\it linearly}) {\it $k$-choosable}. In this paper, we improve some bounds on the 2-frugal choosability and linear choosability of graphs with small maximum average degree.

Domaines: Mathématique discrète [cs.DM]

Nathann Cohen, Frédéric Havet. Linear and 2-frugal choosability of graphs of small maximum average degree. [Research Report] RR-7213, INRIA. 2010. ⟨inria-00459692⟩