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Colorings of Plane Graphs Without Long Monochromatic Facial Paths
Discussiones Mathematicae Graph Theory, 2021 Let G be a plane graph. A facial path of G is a subpath of the boundary walk of a face of G. We prove that each plane graph admits a 3-coloring (a 2-coloring) such that every monochromatic facial path has at most 3 vertices (at most 4 vertices).
Czap Július +2 more
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The list Distinguishing Number Equals the Distinguishing Number for Interval Graphs
Discussiones Mathematicae Graph Theory, 2017 A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring
Immel Poppy, Wenger Paul S.
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On list colouring and list homomorphism of permutation and interval graphs [PDF]
, 2014 List coloring is an NP-complete decision problem even if the total number of colors is three. It is hard even on planar bipartite graphs. We give a polynomial-time algorithm for solving list coloring of permutation graphs with a bounded total number of ...
Tardos, Gábor +2 more
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Tight Lower Bounds for List Edge Coloring [PDF]
, 2018 The fastest algorithms for edge coloring run in time 2^m n^{O(1)}, where m and n are the number of edges and vertices of the input graph, respectively. For dense graphs, this bound becomes 2^{Theta(n^2)}.
Kowalik, Lukasz, Socala, Arkadiusz
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Discrete Mathematics, 2000 Let \(R= \{1,2,\dots, r\}\) be a set of colors, and let \(\ell: V(G)\to 2^R\) be a function. A proper coloring \(c: V(G)\to R\) is said to be a list coloring if \(c(x)\in\ell(x)\) for all \(x\). If there is an integer \(s\) such that all functions \(\ell\) with \(|\ell(x)|= s\) have a list coloring, then \(G\) is said to be \(s\)-choosable. The authors
Michael O. Albertson +2 more
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Planar Graphs with the Distance of 6--Cycles at Least 2 from Each Other Are DP-3-Colorable
Mathematics, 2020 DP-coloring as a generalization of list coloring was introduced by Dvořák and Postle recently. In this paper, we prove that every planar graph in which the distance between 6−-cycles is at least 2 is DP-3-colorable, which extends the result of Yin and Yu
Yueying Zhao, Lianying Miao
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An Analogue of DP-Coloring for Variable Degeneracy and its Applications
Discussiones Mathematicae Graph Theory, 2022 A graph G is list vertex k-arborable if for every k-assignment L, one can choose f(v) ∈ L(v) for each vertex v so that vertices with the same color induce a forest. In [6], Borodin and Ivanova proved that every planar graph without 4-cycles adjacent to 3-
Sittitrai Pongpat, Nakprasit Kittikorn
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List colorings and reducibility
Discrete Applied Mathematics, 1997 Let \(G\) be a simple graph, \(L(v)\) a list of allowed colors assigned to each vertex \(v\) of \(G\), and \(U\) an arbitrary subset of the vertex set. The graph \(G\) is called \(k\)-choosable if, for any list assignment with \(|L(v)|=k\) for all \(v\in V\), it is possible to color all vertices with colors from their lists in a proper way (i.e., no ...
Zsolt Tuza, Margit Voigt
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List-Distinguishing Colorings of Graphs [PDF]
The Electronic Journal of Combinatorics, 2011 A coloring of the vertices of a graph $G$ is said to be distinguishing provided that no nontrivial automorphism of $G$ preserves all of the vertex colors. The distinguishing number of $G$, denoted $D(G)$, is the minimum number of colors in a distinguishing coloring of $G$.
Michael Ferrara +2 more
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Nonrepetitive List Colorings of the Integers [PDF]
Annals of Combinatorics, 2021 AbstractA coloring of the integers is nonrepetitive if no two adjacent intervals have the same color sequence. A beautiful theorem of Thue asserts that there exists a nonrepetitive coloring of $${\mathbb {N}}$$ N using only three colors. We obtain some generalizations of this result in which the adjacency of intervals
Bartłomiej Bosek +3 more
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