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.

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 On the Diameter of Random Planar Graphs

2014 ◽  
Vol 24 (1) ◽  
pp. 145-178 ◽  
Author(s):  
GUILLAUME CHAPUY ◽  
ÉRIC FUSY ◽  
OMER GIMÉNEZ ◽  
MARC NOY
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Formula Group ◽  
Image Position ◽  
Connected Planar Graph

We show that the diameter diam(Gn) of a random labelled connected planar graph withnvertices is equal ton1/4+o(1), in probability. More precisely, there exists a constantc> 0 such that$$ P(\D(G_n)\in(n^{1/4-\e},n^{1/4+\e}))\geq 1-\exp(-n^{c\e}) $$for ε small enough andn ≥ n0(ε). We prove similar statements for 2-connected and 3-connected planar graphs and maps.

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 Tree-width and large grid minors in planar graphs

2011 ◽  
Vol Vol. 13 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Alexander Grigoriev
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Tree Decomposition ◽  
International Audience ◽  
Tree Width ◽  
Grid Minor ◽  
Large Grid

Graphs and Algorithms International audience We show that for a planar graph with no g-grid minor there exists a tree-decomposition of width at most 5g - 6. The proof is constructive and simple. The underlying algorithm for the tree-decomposition runs in O(n(2) log n) time.

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.

A LINEAR-TIME ALGORITHM TO FIND FOUR INDEPENDENT SPANNING TREES IN FOUR CONNECTED PLANAR GRAPHS

1999 ◽  
Vol 10 (02) ◽  
pp. 195-210 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
DAISHIRO TAKAHASHI ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Natural Number ◽  
Planar Graph ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Independent Spanning Trees ◽  
Connected Planar Graph

Given a graph G, a designated vertex r and a natural number k, we wish to find k "independent" spanning trees of G rooted at r, that is, k spanning trees such that the k paths connecting r and any vertex v in the k trees are internally disjoint. In this paper we give a linear-time algorithm to find four independent spanning trees in a 4-connected planar graph.

The random division of faces in a planar graph

1996 ◽  
Vol 28 (2) ◽  
pp. 331-331
Author(s):  
Richard Cowan ◽  
Simone Chen
Keyword(s):  
Planar Graph ◽  
Limiting Distribution ◽  
Face Type ◽  
Connected Planar Graph ◽  
Random Division

Consider a connected planar graph. A bounded face is said to be of type k, or is called a k-face, if the boundary of that face contains k edges. Under various natural rules for randomly dividing bounded faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.

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 Rooted $K_4$-Minors

10.37236/3476 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Ruy Fabila-Monroy ◽  
David R. Wood
Keyword(s):  
Planar Graph ◽  
Pairwise Disjoint ◽  
Single Face ◽  
Special Cases ◽  
Connected Subgraphs ◽  
Connected Planar Graph

Let $a,b,c,d$ be four vertices in a graph $G$. A $K_4$ minor rooted at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.

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