Results 11 to 20 of about 7,403 (166)
On the Dichromatic Number of Surfaces [PDF]
The Electronic Journal of Combinatorics, 2022 In this paper, we give bounds on the dichromatic number $\vec{\chi}(\Sigma)$ of a surface $\Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $\Sigma$. We determine the asymptotic behaviour of $\vec{\chi}(\Sigma)$ by showing that there exist constants $a_1$ and $a_2$ such that, $ a_1\frac{\sqrt{-c}}{\log(-c)} \leq ...
Pierre Aboulker +3 more
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Sharp bounds for Laplacian spectral moments of digraphs with a fixed dichromatic number [PDF]
Discrete Mathematics, 2023 The $k$-th Laplacian spectral moment of a digraph $G$ is defined as $\sum_{i=1}^n λ_i^k$, where $λ_i$ are the eigenvalues of the Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this invariant is better known as the Laplacian energy of $G$.
Xiuwen Yang, Hajo Broersma, Ligong Wang
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Complete minors in digraphs with given dichromatic number [PDF]
, 2021 7 pages ...
Tamás Mészáros, Raphael Steiner
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Uncountable dichromatic number without short directed cycles [PDF]
Journal of Graph Theory, 2019 AbstractHajnal and Erdős proved that a graph with uncountable chromatic number cannot avoid short cycles, it must contain, for example, (among other obligatory subgraphs). It was shown recently by Soukup that, in contrast of the undirected case, it is consistent that for any there exists an uncountably dichromatic digraph without directed cycles ...
Attila Joó
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Dichromatic number and forced subdivisions [PDF]
Journal of Combinatorial Theory, Series B, 2021 24 pages, 1 ...
Lior Gishboliner +2 more
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On the dichromatic number of surfaces [PDF]
, 2021 In this paper, we give bounds on the dichromatic number $\vec{\chi}(\Sigma)$ of a surface $\Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $\Sigma$. We determine the asymptotic behaviour of $\vec{\chi}(\Sigma)$ by showing that there exist constants $a_1$ and $a_2$ such that, $ a_1\frac{\sqrt{-c}}{\log(-c)} \leq \vec{\
P Aboulker +3 more
+5 more sources
On a variant of dichromatic number for digraphs with prescribed sets of arcs [PDF]
, 2023 In this paper, we consider a variant of dichromatic number on digraphs with prescribed sets of arcs. Let $D$ be a digraph and let $Z_1, Z_2$ be two sets of arcs in $D$. For a subdigraph $H$ of $D$, let $A(H)$ denote the set of all arcs of $H$. Let $μ(D, Z_1, Z_2)$ be the minimum number of parts in a vertex partition $\mathcal{P}$ of $D$ such that for ...
O‐joung Kwon, Xiaopan Lian
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$P_{k}$-freeness implies small dichromatic number [PDF]
, 2015 We propose a purely combinatorial quadratic time algorithm that for any $n$-vertex $P_{k}$-free tournament $T$, where $P_{k}$ is a directed path of length $k$, finds in $T$ a transitive subset of order $n^{\frac{c}{k\log(k)^{2}}}$. As a byproduct of our method, we obtain subcubic $O(n^{1-\frac{c}{k\log(k)^{2}}})$-approximation algorithm for the optimal
Krzysztof Choromański
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Chromatic Number and Dichromatic Polynomial of Digraphs [PDF]
, 2017 Let $G$ be a graph of order $n$. It is well-known that $ (G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $ (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 $ (D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1 ...
Saieed Akbari +3 more
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Spectral properties of digraphs with a fixed dichromatic number [PDF]
, 2023 Spectral graph theory is a very important research topic in algebraic graph theory and combinatorial matrix theory. Spectral graph theory primarily utilizes the spectral properties of graph matrices, including eigenvalues (or combinations of eigenvalues) and their associated eigenvectors, to study the structural properties of graphs.
Xiuwen Yang
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