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

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

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.

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.

scholarly journals Face-Degree Bounds for Planar Critical Graphs

10.37236/5895 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Ligang Jin ◽  
Yingli Kang ◽  
Eckhard Steffen
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Upper Bounds ◽  
Partial Characterization ◽  
Critical Graphs ◽  
Degree Bounds

The only remaining case of a well known conjecture of Vizing states that there is no planar graph with maximum degree 6 and edge chromatic number 7. We introduce parameters for planar graphs,  based on the degrees of the faces, and study the question whether there are upper bounds for these parameters for planar edge-chromatic critical graphs. Our results provide upper bounds on these parameters for smallest counterexamples to Vizing's conjecture, thus providing a partial characterization of such graphs, if they exist.For $k \leq 5$ the results give insights into the structure of planar edge-chromatic critical graphs.

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