Cycle Bases Structure of out Planar Graphs on the Projective Plane in Mechanics Engineering

2013 ◽  
Vol 312 ◽  
pp. 745-748
Author(s):  
Mei Xu
Keyword(s):  
Projective Plane ◽  
Planar Graph ◽  
Planar Graphs ◽  
Connected Graphs ◽  
Cycle Bases ◽  
The One ◽  
Outer Planar Graph ◽  
Base Structure

In this paper we investigate the cycle base structure of 2-connected graphs on the projective plane and show the minimum cycle bases of 2-connected outer planar graph G in the case of ew (G) 5. Then give a proof about the one-one property between the minimum cycle bases and the shortest no contractible cycles.

scholarly journals KNOT-INEVITABLE PROJECTIONS OF PLANAR GRAPHS

1996 ◽  
Vol 05 (06) ◽  
pp. 877-883 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
TATSUYA TSUKAMOTO
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Spatial Graph ◽  
Torus Knot ◽  
Nonplanar Graph

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

1969 ◽  
Vol 21 ◽  
pp. 992-1000 ◽  
Author(s):  
L. W. Beineke
Keyword(s):  
Projective Plane ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Graphs ◽  
Minimum Number ◽  
Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

scholarly journals Planar Graphs of Maximum Degree Six without 7-cycles are Class One

10.37236/2589 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Danjun Huang ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

In this paper, we prove that every planar graph of maximum degree six without 7-cycles is class one.

scholarly journals Matchings, Cycle Bases, and the Maximum Genus of a Graph

10.37236/2479 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michal Kotrbčík ◽  
Martin Škoviera
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Perfect Matching ◽  
Intersection Graph ◽  
Matching Number ◽  
Maximum Genus ◽  
Cycle Space ◽  
Intersection Graphs ◽  
Connected Graphs ◽  
Cycle Bases

We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is paid to cycle bases consisting of fundamental cycles with respect to a given spanning tree. For $4$-edge-connected graphs, the intersection graph with respect to any spanning tree (and, in fact, with respect to any basis) has either a perfect matching or a matching missing exactly one vertex. We show that if a graph is not $4$-edge-connected, different spanning trees may lead to intersection graphs with different matching numbers. We also show that there exist $2$-edge connected graphs for which the set of values of matching numbers of their intersection graphs contains arbitrarily large gaps.

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Discrete Math ◽  
Adjacent Vertex ◽  
Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

A Important Property on Planar Graph

2011 ◽  
Vol 50-51 ◽  
pp. 245-248
Author(s):  
Na Na Li ◽  
Shu Jun Wang ◽  
Xian Rui Meng
Keyword(s):  
Planar Graph ◽  
Planar Graphs

In 1994, C.Thomassen proved that every planar graph is (namely ). In 1993, M.Voigt shown that there are planar graphs which are not . But no one know whether every planar graph is . In this paper, we give a important property on planar graph that “Every planar graph is ” and “Every planar graph is free ” are equivalent.

scholarly journals The maximum Wiener index of maximal planar graphs

2020 ◽  
Vol 40 (4) ◽  
pp. 1121-1135
Author(s):  
Debarun Ghosh ◽  
Ervin Győri ◽  
Addisu Paulos ◽  
Nika Salia ◽  
Oscar Zamora
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Planar Graphs ◽  
Wiener Index ◽  
Maximal Planar Graph

Abstract The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n-vertex maximal planar graph is at most $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

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