On Split-Coloring Problems
We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area.
Keywordssplit-coloring vertex covering by split graphs partitioning packing
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- C. Berge, Graphes, Bordas, Paris, 1983.Google Scholar
- G. Dirac, “On rigid circuit graphs,” Abh. Math. Sem. Univ. Hamburg, no. 25, pp. 71–76, 1961.Google Scholar
- S. Földes and P. Hammer, “On split graphs and some related questions,” in Problèmes Combinatoires et Théorie des Graphes, Orsay, France, Colloques Internationnaux C.N.R.S. 260, 1976, pp. 139–140.Google Scholar
- S. Földes and P. Hammer, “Split graphs,” Congressum Numerantium, vol. 19, pp. 311–315, 1977.Google Scholar
- E. Gourdin, M. Labbé, and H. Yaman, “Telecommunication and location,” in Facility location, Springer, Berlin, 2002, pp. 275–305.Google Scholar
- N. Mahadev and U. Peled, Threshold Graphs and Related Topics, Ann. Disc. Mat., North-Holland, vol. 56, 1995.Google Scholar