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- W4300547003 abstract "For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of $H$-colorings. In this paper, we prove several results in this area. First, we find a class of graphs ${mathcal H}$ with the property that for each $H in {mathcal H}$, the $n$-vertex tree that minimizes the number of $H$-colorings is the path $P_n$. We then present a new proof of a theorem of Sidorenko, valid for large $n$, that for every $H$ the star $K_{1,n-1}$ is the $n$-vertex tree that maximizes the number of $H$-colorings. Our proof uses a stability technique which we also use to show that for any non-regular $H$ (and certain regular $H$) the complete bipartite graph $K_{2,n-2}$ maximizes the number of $H$-colorings of $n$-vertex $2$-connected graphs. Finally, we show that the cycle $C_n$ maximizes the number of proper colorings of $n$-vertex $2$-connected graphs." @default.
- W4300547003 created "2022-10-03" @default.
- W4300547003 creator A5041234298 @default.
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- W4300547003 date "2015-06-17" @default.
- W4300547003 modified "2023-10-16" @default.
- W4300547003 title "Extremal H-colorings of trees and 2-connected graphs" @default.
- W4300547003 doi "https://doi.org/10.48550/arxiv.1506.05388" @default.
- W4300547003 hasPublicationYear "2015" @default.
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