Geodesic
A Note on Polynomial Algorithm for Cost Coloring of Bipartite Graphs with Δ ≤ 4
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 3, pp. 885-891

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In the note we consider vertex coloring of a graph in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of coloring is the sum of costs incurred at each vertex. We show that the minimum cost coloring problem for n-vertex bipartite graph of degree Δ ≤ 4 can be solved in O(n2) time. This extends Jansen’s result [K. Jansen, The optimum cost chromatic partition problem, in: Proc. CIAC’97, Lecture Notes in Comput. Sci. 1203 (1997) 25–36] for paths and cycles to subgraphs of biquartic graphs.
Keywords: bipartite graph, chromatic sum, cost coloring, NP-completeness, polynomial algorithm 
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Giaro, Krzysztof; Kubale, Marek. A Note on Polynomial Algorithm for Cost Coloring of Bipartite Graphs with Δ ≤ 4. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 3, pp. 885-891. http://geodesic.mathdoc.fr/item/DMGT_2020_40_3_a12/