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Cornell University Library

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For graphs G and H, a homomorphism from G to H, or H-coloring of G, is a map from the vertices of G to the vertices of H that preserves adjacency. When H is composed of an edge with one looped endvertex, an H-coloring of G corresponds to an independent set in G. Galvin showed that, for sufficiently large n, the complete bipartite graph Kδ,n-δ is the n-vertex graph with minimum degree δ that has the largest number of independent sets.

In this paper, we begin the project of generalizing this result to arbitrary H. Writing hom(G, H) for the number of H-colorings of G, we show that for fixed H and δ = 1 or δ = 2,

hom(G, H) ≤ max{hom(Kδ+1,H)nδ =1, hom(Kδ,δ,H)n, hom(Kδ,n-δ,H)}

for any n-vertex G with minimum degree δ (for sufficiently large n). We also provide examples of H for which the maximum is achieved by hom(Kδ+1, H)nδ+1 and other H for which the maximum is achieved by hom(Kδ,δ,H)n. For δ ≥ 3 (and sufficiently large n), we provide a infinite family of H for which hom(G, H) ≤ hom (Kδ,n-δ, H) for any n-vertex G with minimum degree δ. The results generalize to weighted H-colorings.


Published version. (July 24, 2013). Permalink. © The Author 2013. Used with permission.

John Engbers was affiliated with the University of Notre Dame at the time of publication.