Universal H-colorable graphs without a given configuration

Authors:
Paul A. Dreyer, Jr.;Christopher Malon;Jaroslav Nesetril
Affiliations:
Department of Mathematics, Rutgers University, RAND, 1700 Main Street, P.O. Box 2138,Santa Monica, CA;Department of Mathematics, Massachusetts Institute of Technology;Department of Applied Mathematics, Institute of Theoretical Computer Science(ITI), Charles University, Czech Republic
Venue:
Discrete Mathematics
Year:
2002

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Abstract

For every pair of finite connected graphs F and H, and every positive integer k, we construct a universal graph U with the following properties: (1) There is a homomorphism π: U → H, but no homomorphism from F to U. (2) For every graph G with maximum degree no more than k having a homomorphism h:G → H, but no homomorphism from F to G, there is a homomorphism α: G → U, such that h = π o α. Particularly, this solves a problem presented in [1] and [2] regarding the chromatic number of a universal graph.