Results 31 to 40 of about 192,888 (294)
The Book Thickness of 1-Planar Graphs is Constant [PDF]
Algorithmica, 2016 In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges on the same page do not cross. In this paper, we prove that every $1$-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant ...
Michael A. Bekos +3 more
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The structure of 1-planar graphs
Discrete Mathematics, 2007 AbstractA graph is called 1-planar if it can be drawn in the plane so that each its edge is crossed by at most one other edge. In the paper, we study the existence of subgraphs of bounded degrees in 1-planar graphs. It is shown that each 1-planar graph contains a vertex of degree at most 7; we also prove that each 3-connected 1-planar graph contains an
Igor Fabrici, Tomáš Madaras
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Computational Study on a PTAS for Planar Dominating Set Problem
Algorithms, 2013 The dominating set problem is a core NP-hard problem in combinatorial optimization and graph theory, and has many important applications. Baker [JACM 41,1994] introduces a k-outer planar graph decomposition-based framework for designing polynomial time ...
Qian-Ping Gu, Marjan Marzban
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On total colorings of 1-planar graphs [PDF]
Journal of Combinatorial Optimization, 2013 A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs with maximum degree at least 13.
Guizhen Liu, Jianfeng Hou, Xin Zhang
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1-planar graphs are odd 13-colorable
Discrete Mathematics, 2023 An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $χ_o(G)$, is the minimum number of colors that admits an odd coloring of $G$.
Runrun Liu, Weifan Wang, Gexin Yu
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A note on odd colorings of 1-planar graphs
Discrete Applied Mathematics, 2023 A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petruševski and Škrekovski, who proved that every planar graph admits an odd $9$-coloring; they also conjectured that every planar graph admits an odd $5$-coloring. Shortly after,
Cranston, Daniel W. +2 more
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Structural properties of 1-planar graphs and an application to acyclic edge coloring [PDF]
, 2010 A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e ...
Liu, Guizhen, Wu, Jian-Liang, Zhang, Xin
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Acyclic colouring of 1-planar graphs
Discrete Applied Mathematics, 2001 A graph is said to be 1-planar if it can be embedded into the plane so that each of its edges is crossed by at most one other edge. A coloring of the vertices of a graph is said to be acyclic if every cycle contains at least three colors. The acyclic chromatic number \(a(G)\) of a graph \(G\) is the minimal \(k\) such that \(G\) admits an acyclic \(k\)-
Oleg V. Borodin +3 more
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About Structure of Graph Obstructions for Klein Surface with 9 Vertices
Кібернетика та комп'ютерні технології, 2020 The structure of the 9 vertex obstructive graphs for the nonorientable surface of the genus 2 is established by the method of (-transformations of the graphs.
V.I. Petrenjuk, D.A. Petrenjuk
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