scholarly journals New Upper Bounds for the Heights of Some Light Subgraphs in 1-Planar Graphs with High Minimum Degree

2011 ◽  
Vol Vol. 13 no. 3 (Combinatorics) ◽  
Author(s):  
Xin Zhang ◽  
Jian-Liang Wu ◽  
Guizhen Liu
Keyword(s):  
Planar Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Upper Bounds ◽  
The Family ◽  
International Audience

Combinatorics International audience A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph of minimum degree 6 contains a copy of 4-cycle with all vertices of degree at most 19. In addition, we also show that the complete graph K 4 is light in the family of 1-planar graphs of minimum degree 7, with its height at most 11.

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 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.

Light subgraphs in the family of 1-planar graphs with high minimum degree

2011 ◽  
Vol 28 (6) ◽  
pp. 1155-1168 ◽  
Author(s):  
Xin Zhang ◽  
Gui Zhen Liu ◽  
Jian Liang Wu
Keyword(s):  
Planar Graphs ◽  
Minimum Degree ◽  
The Family

scholarly journals List edge and list total colorings of planar graphs without non-induced 7-cycles

2013 ◽  
Vol Vol. 15 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Aijun Dong ◽  
Guizhen Liu ◽  
Guojun Li
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Total Chromatic Number ◽  
International Audience

Graphs and Algorithms International audience Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.

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.

scholarly journals Induced 2-Degenerate Subgraphs of Triangle-Free Planar Graphs

10.37236/7311 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Tom Kelly
Keyword(s):  
Planar Graph ◽  
Lower Bounds ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Euler’S Formula

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$.  We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph.  We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $\alpha_2(G)$.  We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $\alpha_2(G) \geq \frac{6n - m - 1}{5}$.  By Euler's Formula, this implies $\alpha_2(G) \geq \frac{4}{5}n$.  We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most three, then $\alpha_2(G) \geq \frac{7}{8}n - 18 n_3$.

On Partitioning Planar Graphs

1968 ◽  
Vol 11 (2) ◽  
pp. 203-211 ◽  
Author(s):  
Stephen Hedetniemi
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Single Point ◽  
Upper Bounds ◽  
Point Set

In 1879 Kempe [5] presented what has become the most famous of all incorrect proofs of the Four Colour Conjecture, but even though his proof was erroneous his method has become quite useful. In 1890 Heawood [4] was able to modify Kempe's method to establish the Five Colour Theorem for planar graphs. In this article we show that other modifications of Kempe's method can be made which enable one to establish more results about planar graphs. By this process we obtain upper bounds for several parameters which involve partitioning the point set of a graph. In particular, we show that the point set of any planar graph can be partitioned into four or less subsets such that the subgraph induced by each subset is either disconnected or trivial (consists of a single point). We also show that the point set of any planar graph can be partitioned into three or less subsets such that the subgraph induced by each subset contains no cycles.

The structure of plane graphs with independent crossings and its applications to coloring problems

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Xin Zhang ◽  
Guizhen Liu
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Plane Graph ◽  
Plane Graphs ◽  
Total Chromatic Number ◽  
Linear Arboricity

AbstractIf a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ℈Δ/2⌊ if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.

scholarly journals Toughness of $K_{a,t}$-Minor-free Graphs

10.37236/635 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guantao Chen ◽  
Yoshimi Egawa ◽  
Ken-ichi Kawarabayashi ◽  
Bojan Mohar ◽  
Katsuhiro Ota
Keyword(s):  
Planar Graph ◽  
Positive Constant ◽  
Complete Graph ◽  
Planar Graphs ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Free Graph ◽  
Minor Free Graphs ◽  
Vertex Sets ◽  
Connected Planar Graph

The toughness of a non-complete graph $G$ is the minimum value of $\frac{|S|}{\omega(G-S)}$ among all separating vertex sets $S\subset V(G)$, where $\omega(G-S)\ge 2$ is the number of components of $G-S$. It is well-known that every $3$-connected planar graph has toughness greater than $1/2$. Related to this property, every $3$-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a $2$-walk, etc. Realizing that 3-connected planar graphs are essentially the same as 3-connected $K_{3,3}$-minor-free graphs, we consider a generalization to $a$-connected $K_{a,t}$-minor-free graphs, where $3\le a\le t$. We prove that there exists a positive constant $h(a,t)$ such that every $a$-connected $K_{a,t}$-minor-free graph $G$ has toughness at least $h(a,t)$. For the case where $a=3$ and $t$ is odd, we obtain the best possible value for $h(3,t)$. As a corollary it is proved that every such graph of order $n$ contains a cycle of length $\Omega(\log_{h(a,t)} n)$.

scholarly journals On the diameter of random planar graphs

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Guillaume Chapuy ◽  
Eric Fusy ◽  
Omer Gimenez ◽  
Marc Noy
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
International Audience ◽  
Connected Planar Graph

International audience We show that the diameter $D(G_n)$ of a random (unembedded) labelled connected planar graph with $n$ vertices is asymptotically almost surely of order $n^{1/4}$, in the sense that there exists a constant $c>0$ such that $P(D(G_n) \in (n^{1/4-\epsilon} ,n^{1/4+\epsilon})) \geq 1-\exp (-n^{c\epsilon})$ for $\epsilon$ small enough and $n$ large enough $(n \geq n_0(\epsilon))$. We prove similar statements for rooted $2$-connected and $3$-connected embedded (maps) and unembedded planar graphs.

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