scholarly journals Erdös-Gyárfás Conjecture for Cubic Planar Graphs

10.37236/3252 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Christopher Carl Heckman ◽  
Roi Krakovski
Keyword(s):  
Planar Graphs ◽  
Minimum Degree ◽  
Negative Integer ◽  
Simple Cycle ◽  
Discharging Method ◽  
Computer Based

In 1995, Paul Erdös and András Gyárfás conjectured that for every graph of minimum degree at least 3, there exists a non-negative integer $m$ such that $G$ contains a simple cycle of length $2^m$. In this paper, we prove that the conjecture holds for 3-connected cubic planar graphs. The proof is long, computer-based in parts, and employs the Discharging Method in a novel way.

scholarly journals Finding small simple cycle separators for 2-connected planar graphs

1986 ◽  
Vol 32 (3) ◽  
pp. 265-279 ◽  
Author(s):  
Gary L. Miller
Keyword(s):  
Planar Graphs ◽  
Simple Cycle

Transformations for maximal planar graphs with minimum degree five

1995 ◽  
pp. 366-371
Author(s):  
Jean Hardouin Duparc ◽  
Philippe Rolland
Keyword(s):  
Planar Graphs ◽  
Minimum Degree

Light Edges in 3-Connected 2-Planar Graphs With Prescribed Minimum Degree

2016 ◽  
Vol 41 (3) ◽  
pp. 1265-1274
Author(s):  
Zai Ping Lu ◽  
Ning Song
Keyword(s):  
Planar Graphs ◽  
Minimum Degree

scholarly journals The 7-cycle C7 is light in the family of planar graphs with minimum degree 5

2007 ◽  
Vol 307 (11-12) ◽  
pp. 1430-1435 ◽  
Author(s):  
T. Madaras ◽  
R. Škrekovski ◽  
H.-J. Voss
Keyword(s):  
Planar Graphs ◽  
Minimum Degree ◽  
The Family

scholarly journals The Positive Minimum Degree Game on Sparse Graphs

10.37236/1174 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
József Balogh ◽  
András Pluhár
Keyword(s):  
Complete Graph ◽  
Special Form ◽  
Minimum Degree ◽  
Sharp Bound ◽  
Sparse Graphs ◽  
Discharging Method ◽  
Graph Game

In this note we investigate a special form of degree games defined by D. Hefetz, M. Krivelevich, M. Stojaković and T. Szabó. Usually the board of a graph game is the edge set of $K_n$, the complete graph on $n$ vertices. Maker and Breaker alternately claim an edge, and Maker wins if his edges form a subgraph with prescribed properties; here a certain minimum degree. In the special form the board is no longer the whole edge set of $K_n$, Maker first selects as few edges of $K_n$ as possible in order to win, and our goal is to compute the necessary size of that board. Solving a question of Hefetz et al., we show, using the discharging method, that the sharp bound is around $10n/7$ for the positive minimum degree game.

scholarly journals The structure and the list 3-dynamic coloring of outer-1-planar graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Local Structure ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Outer Region ◽  
The Other ◽  
Structural Theorem ◽  
Other Hand

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

scholarly journals A note on 1-planar graphs with minimum degree 7

2021 ◽  
Vol 289 ◽  
pp. 230-232
Author(s):  
Therese Biedl
Keyword(s):  
Planar Graphs ◽  
Minimum Degree

Short and Simple Cycle Separators in Planar Graphs

2013 ◽  
pp. 26-40 ◽  
Author(s):  
Eli Fox-Epstein ◽  
Shay Mozes ◽  
Phitchaya Mangpo Phothilimthana ◽  
Christian Sommer
Keyword(s):  
Planar Graphs ◽  
Simple Cycle

On Light Edges and Triangles in Planar Graphs of Minimum Degree Five

2006 ◽  
Vol 170 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Oleg V. Borodin ◽  
Daniel P. Sanders
Keyword(s):  
Planar Graphs ◽  
Minimum Degree

Light edges in 1-planar graphs with prescribed minimum degree

2012 ◽  
Vol 32 (3) ◽  
pp. 545 ◽  
Author(s):  
Dávid Hudák ◽  
Peter Šugerek
Keyword(s):  
Planar Graphs ◽  
Minimum Degree
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