Rapport Année : 2025

Backbone Colouring of Chordal Graphs

Résumé

A proper k-colouring of a graph G = (V, E) is a function c : V (G) → {1, . . . , k} such that c(u) ̸ = c(v) for every edge uv ∈ 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 |c(u) -c(v)| ⩾ q for every edge uv ∈ E(H). The q-backbone chromatic number BBCq(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.

[1] ask whether, for any chordal graph G and any spanning forest H of G, we have that BBC2(G, H) ⩽ 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 BBC2(G, H) ⩾ 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 BBC2(G, H) ⩽ χ(G) + O( χ(G)). We finally show that BBC2(G, H) ⩽ 3 2 χ(G) + O(1) holds whenever G is chordal and H is C4-free.

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Dates et versions

hal-05058457 , version 1 (06-05-2025)
hal-05058457 , version 2 (04-08-2025)

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  • HAL Id : hal-05058457 , version 2

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Júlio Araújo, Nicolas Nisse, Lucas Picasarri-Arrieta. Backbone Colouring of Chordal Graphs. Inria. 2025. ⟨hal-05058457v2⟩
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