scholarly journals Characterization of Cycle Obstruction Sets for Improper Coloring Planar Graphs

2018 ◽  
Vol 32 (2) ◽  
pp. 1209-1228 ◽  
Author(s):  
Ilkyoo Choi ◽  
Chun-Hung Liu ◽  
Sang-il Oum
Keyword(s):  
Planar Graphs ◽  
Improper Coloring

Note on improper coloring of $1$-planar graphs

2019 ◽  
Vol 69 (4) ◽  
pp. 955-968
Author(s):  
Yanan Chu ◽  
Lei Sun ◽  
Jun Yue
Keyword(s):  
Planar Graphs ◽  
Improper Coloring

A Categorical Characterization of the Four Colour Theorem

1986 ◽  
Vol 29 (4) ◽  
pp. 426-431 ◽  
Author(s):  
Barry Fawcett
Keyword(s):  
Planar Graphs ◽  
Edge Preserving

AbstractThe surjectivity of epimorphisms in the category of planar graphs and edge-preserving maps follows from and is implied by the Four Colour Theorem. The argument that establishes the equivalence is not combinatorially complex.

scholarly journals A characterization of level planar graphs

2004 ◽  
Vol 280 (1-3) ◽  
pp. 51-63 ◽  
Author(s):  
Patrick Healy ◽  
Ago Kuusik ◽  
Sebastian Leipert
Keyword(s):  
Planar Graphs

A Characterization of Planar Graphs by Pseudo-Line Arrangements

Algorithmica ◽  
2003 ◽  
Vol 35 (3) ◽  
pp. 269-285 ◽  
Author(s):  
Hisao Tamaki ◽  
Takeshi Tokuyama
Keyword(s):  
Planar Graphs ◽  
Line Arrangements

A Characterization of Almost-Planar Graphs

1996 ◽  
Vol 5 (3) ◽  
pp. 227-245 ◽  
Author(s):  
Bradley S. Gubser
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Planar Graphs ◽  
Explicit Description ◽  
Famous Result ◽  
Minor Criterion ◽  
Excluded Minor

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

A characterization of signed planar graphs with rank at most 4

2015 ◽  
Vol 64 (5) ◽  
pp. 807-817 ◽  
Author(s):  
Fenglei Tian ◽  
Dengyin Wang ◽  
Min Zhu
Keyword(s):  
Planar Graphs

scholarly journals Further characterizations and Helly-property in k-trees

2016 ◽  
Vol 15 (3) ◽  
pp. 1-8
Author(s):  
H P Patil
Keyword(s):  
Planar Graphs ◽  
Similar Type ◽  
The Other ◽  
Forbidden Subgraphs ◽  
Outerplanar Graphs ◽  
Helly Property ◽  
Maximal Outerplanar Graphs

The purpose of this paper is to obtain a characterization of $k$-trees in terms of $k$-connectivity and forbidden subgraphs. Also, we present the other characterizations of $k$-trees containing the full vertices by using the join operation. Further, we establish the property of $k$-trees dealing with the degrees and formulate the Helly-property for a family of nontrivial $k$-paths in a $k$-tree. We study the planarity of $k$-trees and express the maximal outerplanar graphs in terms of 2-trees and $K_2$-neighbourhoods. Finally, the similar type of results for the maximal planar graphs are obtained.

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