On sparse graphs with given colorings and homomorphisms

  • Authors:

  • Jaroslav Nešetřil;Xuding Zhu

  • Affiliations:

  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Institute for Theoretical Computer Science (ITI), Charles University, Malostranské nám.25, 11800 Praha 1, Czech Rep ...;Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424 Taiwan

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2004

Quantified Score

Hi-index 0.00

Visualization

Abstract

We prove that for every graph H and positive integers k and l there exists a graph G with girth at least l such that for all graphs H' with at most k vertices there exists a homomorphism G → H' if and only if there exists a homomorphism H → H'. This implies (for H = Kk) the classical result of Erdös and other generalizations (such as Sparse Incomparability Lemma). We refine the above statement to the 1-1 correspondence between the set of all homomorphisms G → H' and the set of all homomorphisms H → H'. This in turn yields the existence of sparse uniquely H-colorable graphs and, perhaps surprisingly, provides a characterization of the graphs H for which the analog of Müller's theorem holds for H-colorings.