Backbone colouring of chordal graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918114447327232 |
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| author | Araújo, Júlio Nisse, Nicolas Picasarri-Arrieta, Lucas |
| author_facet | Araújo, Júlio Nisse, Nicolas Picasarri-Arrieta, Lucas |
| contents | A proper $k$-colouring of a graph $G=(V,E)$ is a function $c: V(G)\to \{1,\ldots,k\}$ such that $c(u)\neq c(v)$ for every edge $uv\in E(G)$. The chromatic number $χ(G)$ is the minimum $k$ such that there exists a proper $k$-colouring of $G$. Given a spanning subgraph $H$ of $G$, a $q$-backbone $k$-colouring of $(G,H)$ is a proper $k$-colouring $c$ of $G$ such that $\lvert c(u)-c(v)\rvert \ge q$ for every edge $uv\in E(H)$. The $q$-backbone chromatic number ${\rm BBC}_q(G,H)$ is the smallest $k$ for which there exists a $q$-backbone $k$-colouring of $(G,H)$. In their seminal paper, Broersma et al.~\cite{BFGW07} ask whether, for any chordal graph $G$ and any spanning forest $H$ of $G$, we have that ${\rm BBC}_2(G,H)\leq χ(G)+O(1)$.
In this work, we first show that this is true as long as $H$ is bipartite and $G$ is an interval graph in which each vertex belongs to at most two maximal cliques. We then show that this does not extend to bipartite graphs as backbone by exhibiting a family of chordal graphs $G$ with spanning bipartite subgraphs $H$ satisfying ${\rm BBC}_2(G,H)\geq \frac{5χ(G)}{3}$. Then, we show that if $G$ is chordal and $H$ has bounded maximum average degree (in particular, if $H$ is a forest), then ${\rm BBC}_2(G,H)\leq χ(G)+O(\sqrt{χ(G)})$. We finally show that ${\rm BBC}_2(G,H)\leq \frac{3}{2}χ(G)+O(1)$ holds whenever $G$ is chordal and $H$ is $C_4$-free. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Backbone colouring of chordal graphs Araújo, Júlio Nisse, Nicolas Picasarri-Arrieta, Lucas Combinatorics Discrete Mathematics A proper $k$-colouring of a graph $G=(V,E)$ is a function $c: V(G)\to \{1,\ldots,k\}$ such that $c(u)\neq c(v)$ for every edge $uv\in E(G)$. The chromatic number $χ(G)$ is the minimum $k$ such that there exists a proper $k$-colouring of $G$. Given a spanning subgraph $H$ of $G$, a $q$-backbone $k$-colouring of $(G,H)$ is a proper $k$-colouring $c$ of $G$ such that $\lvert c(u)-c(v)\rvert \ge q$ for every edge $uv\in E(H)$. The $q$-backbone chromatic number ${\rm BBC}_q(G,H)$ is the smallest $k$ for which there exists a $q$-backbone $k$-colouring of $(G,H)$. In their seminal paper, Broersma et al.~\cite{BFGW07} ask whether, for any chordal graph $G$ and any spanning forest $H$ of $G$, we have that ${\rm BBC}_2(G,H)\leq χ(G)+O(1)$. In this work, we first show that this is true as long as $H$ is bipartite and $G$ is an interval graph in which each vertex belongs to at most two maximal cliques. We then show that this does not extend to bipartite graphs as backbone by exhibiting a family of chordal graphs $G$ with spanning bipartite subgraphs $H$ satisfying ${\rm BBC}_2(G,H)\geq \frac{5χ(G)}{3}$. Then, we show that if $G$ is chordal and $H$ has bounded maximum average degree (in particular, if $H$ is a forest), then ${\rm BBC}_2(G,H)\leq χ(G)+O(\sqrt{χ(G)})$. We finally show that ${\rm BBC}_2(G,H)\leq \frac{3}{2}χ(G)+O(1)$ holds whenever $G$ is chordal and $H$ is $C_4$-free. |
| title | Backbone colouring of chordal graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2508.02980 |