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- W4297749501 abstract "Let $G$ be a graph of order $n$. It is well-known that $alpha(G)geq sum_{i=1}^n frac{1}{1+d_i}$, where $alpha(G)$ is the independence number of $G$ and $d_1,ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ alpha(D)geq sum_{i=1}^n left( frac{1}{1+d_i^+} + frac{1}{1+d_i^-} - frac{1}{1+d_i}right)$, where $alpha(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $chi(D)leq lceil frac{4k}{5} rceil+2$, where $k=max(Delta^+(D),Delta^-(D))$. We give a short and simple proof for this result. Next, we investigate the chromatic number of tournaments and determine the unique tournament such that for every integer $k>1$, the number of proper $k$-colorings of that tournament is maximum among all strongly connected tournaments with the same number of vertices. Also, we find the chromatic polynomial of the strongly connected tournament with the minimum number of cycles." @default.
- W4297749501 created "2022-09-30" @default.
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- W4297749501 date "2017-11-16" @default.
- W4297749501 modified "2023-09-27" @default.
- W4297749501 title "Chromatic Number and Dichromatic Polynomial of Digraphs" @default.
- W4297749501 doi "https://doi.org/10.48550/arxiv.1711.06293" @default.
- W4297749501 hasPublicationYear "2017" @default.
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