409 research outputs found
Extremal H-Colorings of Graphs with Fixed Minimum Degree
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)n⁄δ =1, hom(Kδ,δ,H)n⁄2δ, 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)n⁄δ+1 and other H for which the maximum is achieved by hom(Kδ,δ,H)n⁄2δ. 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
Impartial coloring games
Coloring games are combinatorial games where the players alternate painting
uncolored vertices of a graph one of colors. Each different ruleset
specifies that game's coloring constraints. This paper investigates six
impartial rulesets (five new), derived from previously-studied graph coloring
schemes, including proper map coloring, oriented coloring, 2-distance coloring,
weak coloring, and sequential coloring. For each, we study the outcome classes
for special cases and general computational complexity. In some cases we pay
special attention to the Grundy function
Online and quasi-online colorings of wedges and intervals
We consider proper online colorings of hypergraphs defined by geometric
regions. We prove that there is an online coloring algorithm that colors
intervals of the real line using colors such that for every
point , contained in at least intervals, not all the intervals
containing have the same color. We also prove the corresponding result
about online coloring a family of wedges (quadrants) in the plane that are the
translates of a given fixed wedge. These results contrast the results of the
first and third author showing that in the quasi-online setting 12 colors are
enough to color wedges (independent of and ). We also consider
quasi-online coloring of intervals. In all cases we present efficient coloring
algorithms
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