H-colouring bipartite graphs

  • Authors:

  • John Engbers;David Galvin

  • Affiliations:

  • Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States;Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

For graphs G and H, an H-colouring of G (or homomorphism from G to H) is a function from the vertices of G to the vertices of H that preserves adjacency. H-colourings generalize such graph theory notions as proper colourings and independent sets. For a given H, k@?V(H) and G we consider the proportion of vertices of G that get mapped to k in a uniformly chosen H-colouring of G. Our main result concerns this quantity when G is regular and bipartite. We find numbers 0=