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.

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.

A counter-example to ‘Wagner's conjecture’ for infinite graphs

1988 ◽  
Vol 103 (1) ◽  
pp. 55-57 ◽  
Author(s):  
Robin Thomas
Keyword(s):  
Graph Theory ◽  
Planar Graphs ◽  
Infinite Sequence ◽  
Infinite Graphs ◽  
Famous Theorem ◽  
Finite Graphs ◽  
Counter Example ◽  
Excluded Minor ◽  
A Minor ◽  
Forbidden Minors

Wagner made the conjecture that given an infinite sequence G1, G2, … of finite graphs there are indices i < j such that Gi is a minor of Gj. (A graph is a minor of another if the first can be obtained by contraction from a subgraph of the second.) The importance of this conjecture is that it yields excluded minor theorems in graph theory, where by an excluded minor theorem we mean a result asserting that a graph possesses a specified property if and only if none of its minors belongs to a finite list of ‘forbidden minors’. A widely known example of an excluded minor theorem is Kuratowski's famous theorem on planar graphs; one of its formulations says that a graph is planar if and only if it has neither K5 nor K3, 3 as a minor. But several other excluded minor theorems have been discovered by now (see e.g. [7–9]).

scholarly journals Extremal $H$-Free Planar Graphs

10.37236/8255 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Yongxin Lan ◽  
Yongtang Shi ◽  
Zi-Xia Song
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Sufficient Conditions ◽  
Subcubic Graph ◽  
Stone Theorem

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory  83 (2016) 213–230], that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield  $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem.  We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to  $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

scholarly journals On a characterization of planar graphs

1981 ◽  
Vol 24 (2) ◽  
pp. 289-294 ◽  
Author(s):  
C. C. Chen
Keyword(s):  
Planar Graph ◽  
Simple Proof ◽  
Planar Graphs

By introducing the concept of a polygon-extension of a planar graph, we provide a simple proof that a graph is planar if and only if every strict elegant ring in the graph is even.

Characterization of the anterior cingulate's role in the at-risk mental state using graph theory

NeuroImage ◽  
2011 ◽  
Vol 56 (3) ◽  
pp. 1531-1539 ◽  
Author(s):  
Louis-David Lord ◽  
Paul Allen ◽  
Paul Expert ◽  
Oliver Howes ◽  
Renaud Lambiotte ◽  
...  
Keyword(s):  
At Risk ◽  
Graph Theory ◽  
Mental State ◽  
At Risk Mental State

Characterization of Decentralized and Quotient Fixed Modes via Graph Theory

2007 ◽  
Author(s):  
Javad Lavaei ◽  
Amir G. Aghdam
Keyword(s):  
Graph Theory

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].

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 p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

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