Geodesic
Optimal Backbone Coloring of Split Graphs with Matching Backbones
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 157-169

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For a graph G with a given subgraph H, the backbone coloring is defined as the mapping c : V(G) → ℕ_+ such that |c(u) − c(v)| ≥ 2 for each edge u, v ∈ E(H) and |c(u) − c(v)| ≥ 1 for each edge u, v ∈ E(G). The backbone chromatic number BBC(G,H) is the smallest integer k such that there exists a backbone coloring with max_v∈V(G) c(v) = k. In this paper, we present the algorithm for the backbone coloring of split graphs with matching backbone.
Keywords: backbone coloring, split graphs, matching 
@article{DMGT_2015_35_1_a12,
     author = {Turowski, Krzysztof},
     title = {Optimal {Backbone} {Coloring} of {Split} {Graphs} with {Matching} {Backbones}},
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Turowski, Krzysztof. Optimal Backbone Coloring of Split Graphs with Matching Backbones. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 157-169. http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a12/