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Discussiones Mathematicae Graph Theory, 2001 Let \(\tau(G)\) denote the number of vertices in a longest path of a graph \(G\). The \(n\)th detour number \(\chi_n(G)\) of a graph \(G\) is the minimum number of colours required to colour the vertices of \(G\) such that no path with more than \(n\) vertices is monocoloured. It is shown that the path partition conjecture, formulated by P. Mihók (see \
Frick, Marietjie, Bullock, Frank
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The Distinguishing Chromatic Number [PDF]
The Electronic Journal of Combinatorics, 2006 In this paper we define and study the distinguishing chromatic number, $\chi_D(G)$, of a graph $G$, building on the work of Albertson and Collins who studied the distinguishing number. We find $\chi_D(G)$ for various families of graphs and characterize those graphs with $\chi_D(G)$ $ = |V(G)|$, and those trees with the maximum chromatic distingushing ...
Collins, Karen L., Trenk, Ann N.
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On the local distinguishing chromatic number
AKCE International Journal of Graphs and Combinatorics, 2019 The distinguishing number of graphs is generalized in two directions by Cheng and Cowen (local distinguishing number) and Collins and Trenk (Distinguishing chromatic number). In this paper, we define and study the local distinguishing chromatic number of
Omid Khormali
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Chromatic Ramsey number of acyclic hypergraphs [PDF]
, 2015 Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $\chi(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph are colored with
Gyárfás, András +2 more
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Graphs with tiny vector chromatic numbers and huge chromatic numbers [PDF]
The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., 2003 Summary: \textit{D. Karger, R. Motwani} and \textit{M. Sudan} [J. ACM 45, 246--265 (1998; Zbl 0904.68116)] introduced the notion of a vector coloring of a graph. In particular, they showed that every \(k\)-colorable graph is also vector \(k\)-colorable, and that for constant \(k\), graphs that are vector \(k\)-colorable can be colored by roughly ...
Feige, Uriel +2 more
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On the Locating Chromatic Number of Barbell Shadow Path Graph
Indonesian Journal of Combinatorics, 2021 The locating-chromatic number was introduced by Chartrand in 2002. The locating chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph.
A. Asmiati +2 more
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Chromatic Number and Neutrosophic Chromatic Number
, 2021 New setting is introduced to study chromatic number. Neutrosophic chromatic number and chromatic number are proposed in this way, some results are obtained. Classes of neutrosophic graphs are used to obtains these numbers and the representatives of the colors. Using colors to assigns to the vertices of neutrosophic graphs is applied. Some questions and
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Chromatic-Choosability of Hypergraphs with High Chromatic Number [PDF]
The Electronic Journal of Combinatorics, 2019 It was conjectured by Ohba and confirmed by Noel, Reed and Wu that, for any graph $G$, if $|V(G)|\le 2\chi(G)+1$ then $G$ is chromatic-choosable; i.e., it satisfies $\chi_l(G)=\chi(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We observe that this is also the case for uniform hypergraphs and further propose a
Wang, Wei, Qian, Jianguo
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Improved Hardness of Approximating Chromatic Number [PDF]
, 2013 We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2^{K^{1/3}} colors. This improves the previous result of K versus K^{O(log K)} in Khot [14]
Huang, Sangxia
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Total dominator chromatic number of Kneser graphs
AKCE International Journal of Graphs and Combinatorics, 2023 Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures. A total dominator coloring (briefly, a TDC) of a graph G is a proper coloring (i.e.
Parvin Jalilolghadr, Ali Behtoei
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