Results 11 to 20 of about 380,403 (285)
Acyclic Chromatic Index of 1-Planar Graphs [PDF]
Mathematics, 2022 The acyclic chromatic index χa′(G) of a graph G is the smallest k for which G is a proper edge colorable using k colors. A 1-planar graph is a graph that can be drawn in plane such that every edge is crossed by at most one other edge.
Wanshun Yang +5 more
doaj +2 more sources
On 4-Map Graphs and 1-Planar Graphs and their Recognition Problem [PDF]
arXiv, 2015 We establish a one-to-one correspondence between 1-planar graphs and general and hole-free 4-map graphs and show that 1-planar graphs can be recognized in polynomial time if they are crossing-augmented, fully triangulated, and maximal 1-planar, respectively, with a polynomial of degree 120, 3, and 5, respectively.
Franz J. Brandenburg
arxiv +3 more sources
On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs
Discussiones Mathematicae Graph Theory, 2016 A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets.
Czap Július +2 more
doaj +2 more sources
A note on 1-planar graphs with minimum degree 7 [PDF]
arXiv, 2019 It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
Thérèse Biedl
arxiv +3 more sources
Edge Coloring of Triangle-Free 1-Planar Graphs [PDF]
arXiv, 2010 it is shown that each triangle-free 1-planar graph with maximum degree $\Delta\geq7$ can be $\Delta$-colorable by Discharging Method.
Xin Zhang, Guizhen Liu, Jianliang Wu
arxiv +3 more sources
Recognizing Optimal 1-Planar Graphs in Linear Time [PDF]
Algorithmica, 2016 A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n-8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our algorithm implements a graph reduction system with two rules, which can be used to reduce every optimal 1-planar
F. Brandenburg
arxiv +2 more sources
Joins of 1-planar graphs [PDF]
Acta. Math. Sin.-English Ser. 30 (2014) 1867-1876, 2014 A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join $G+H$ is 1-planar if and only if the pair $[G,H]$ is subgraph-majorized (that is, both $G$ and $H$ are subgraphs of graphs of the major pair) by one of pairs $[C_3 \cup C_3,
J. Czap, Dávid Hudák, T. Madaras
arxiv +3 more sources
Drawing Subcubic 1-Planar Graphs with Few Bends, Few Slopes, and Large Angles [PDF]
arXiv, 2018 We show that the 1-planar slope number of 3-connected cubic 1-planar graphs is at most 4 when edges are drawn as polygonal curves with at most 1 bend each. This bound is obtained by drawings whose vertex and crossing resolution is at least $\pi/4$. On the other hand, if the embedding is fixed, then there is a 3-connected cubic 1-planar graph that needs
Philipp Kindermann +3 more
arxiv +2 more sources
The strong chromatic index of 1-planar graphs [PDF]
Discrete Mathematics & Theoretical Computer ScienceThe chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors.
Yiqiao Wang +3 more
doaj +2 more sources
A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs [PDF]
International Workshop on Combinatorial Algorithms, 2018 We propose a fixed-parameter tractable algorithm for the \textsc{Max-Cut} problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge.
Christine Dahn +2 more
arxiv +2 more sources