Backbone colouring: tree backbones with small diameter in planar graphs

Victor Campos 1 Frédéric Havet 2 Rudini Sampaio 1 Ana Silva 1
2 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : Given a graph $G$ and a spanning subgraph $T$ of $G$, a {\it backbone $k$-colouring} for $(G,T)$ is a mapping $c:V(G)\to\{1,\ldots,k\}$ such that $|c(u)-c(v)|\geq 2$ for every edge $uv\in E(T)$ and $|c(u)-c(v)|\geq 1$ for every edge $uv\in E(G)\setminus E(T)$. The {\it backbone chromatic number} $BBC(G,T)$ is the smallest integer $k$ such that there exists a backbone $k$-colouring of $(G,T)$. In 2007, Broersma et al. \cite{BFG+07} conjectured that $BBC(G,T)\leq 6$ for every planar graph $G$ and every spanning tree $T$ of $G$. In this paper, we prove this conjecture when $T$ has diameter at most four.
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Victor Campos, Frédéric Havet, Rudini Sampaio, Ana Silva. Backbone colouring: tree backbones with small diameter in planar graphs. [Research Report] RR-8151, INRIA. 2012. ⟨hal-00758548⟩

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