scholarly journals The 2-surviving rate of planar graphs with average degree lower than 92

2018 ◽  
Vol 89 (3) ◽  
pp. 341-349
Author(s):  
Przemysław Gordinowicz
Keyword(s):  
Planar Graphs ◽  
Average Degree ◽  
Surviving Rate

scholarly journals Partitioning Sparse Graphs into an Independent Set and a Forest of Bounded Degree

10.37236/6815 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
François Dross ◽  
Mickael Montassier ◽  
Alexandre Pinlou
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Independent Set ◽  
Average Degree ◽  
Bounded Degree ◽  
Sparse Graphs ◽  
Maximum Average Degree

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition. It follows that planar graphs with girth at least $7$ (resp. $8$, $10$) admit an $({\cal I},{\cal F}_5)$-partition (resp. $({\cal I},{\cal F}_3)$-partition, $({\cal I},{\cal F}_2)$-partition).

scholarly journals The surviving rate of planar graphs

2012 ◽  
Vol 416 ◽  
pp. 65-70 ◽  
Author(s):  
Jiangxu Kong ◽  
Weifan Wang ◽  
Xuding Zhu
Keyword(s):  
Planar Graphs ◽  
Surviving Rate

Structural properties and surviving rate of planar graphs

2014 ◽  
Vol 06 (04) ◽  
pp. 1450052 ◽  
Author(s):  
Jiangxu Kong ◽  
Lianzhu Zhang ◽  
Weifan Wang
Keyword(s):  
Structural Properties ◽  
Connected Graph ◽  
Planar Graphs ◽  
Time Interval ◽  
Average Proportion ◽  
Surviving Rate

Let G be a connected graph. Suppose that a fire breaks out at some vertex. A firefighter starts to protect vertices. At each time interval, the firefighter protects k vertices not yet on fire. At the end of each time interval, the fire spreads to all the unprotected vertices that have a neighbor on fire. The k-surviving rate ρk(G) of G is the average proportion of saved vertices, if the starting vertex of the fire is chosen uniformly at random. A graph G is called k-good if there is a constant c > 0 such that ρk(G) ≥ c. We study structural properties of planar graphs and show that planar graphs are 3-good.

RANDOM PLANAR GRAPHS WITH GIVEN AVERAGE DEGREE

2007 ◽  
pp. 83-102 ◽  
Author(s):  
Stefanie Gerke ◽  
Colin McDiarmid ◽  
Angelika Steger ◽  
Andreas Weißl
Keyword(s):  
Planar Graphs ◽  
Average Degree

scholarly journals The 2-surviving rate of planar graphs without 4-cycles

2012 ◽  
Vol 457 ◽  
pp. 158-165 ◽  
Author(s):  
Weifan Wang ◽  
Jiangxu Kong ◽  
Lianzhu Zhang
Keyword(s):  
Planar Graphs ◽  
Surviving Rate

The 2-surviving rate of planar graphs without 5-cycles

2015 ◽  
Vol 31 (4) ◽  
pp. 1479-1492 ◽  
Author(s):  
Tingting Wu ◽  
Jiangxu Kong ◽  
Weifan Wang
Keyword(s):  
Planar Graphs ◽  
Surviving Rate

scholarly journals The 2-surviving rate of planar graphs without 6-cycles

2014 ◽  
Vol 518 ◽  
pp. 22-31 ◽  
Author(s):  
Weifan Wang ◽  
Stephen Finbow ◽  
Jiangxu Kong
Keyword(s):  
Planar Graphs ◽  
Surviving Rate

scholarly journals 8-star-choosability of a graph with maximum average degree less than 3

2011 ◽  
Vol Vol. 13 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Min Chen ◽  
André Raspaud ◽  
Weifan Wang
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Average Degree ◽  
Vertex Coloring ◽  
Maximum Average Degree ◽  
International Audience ◽  
Star Coloring ◽  
List Chromatic Number ◽  
Proper Vertex

Graphs and Algorithms International audience A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].

scholarly journals On randomly colouring locally sparse graphs

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Alan Frieze ◽  
Juan Vera
Keyword(s):  
Random Graphs ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Glauber Dynamics ◽  
Average Degree ◽  
Sparse Graphs ◽  
International Audience ◽  
General Graphs

International audience We consider the problem of generating a random q-colouring of a graph G=(V,E). We consider the simple Glauber Dynamics chain. We show that if for all v ∈ V the average degree of the subgraph H_v induced by the neighbours of v ∈ V is #x226a Δ where Δ is the maximum degree and Δ >c_1\ln n then for sufficiently large c_1, this chain mixes rapidly provided q/Δ >α , where α #x2248 1.763 is the root of α = e^\1/α \. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs G_\n,p\ with p #x226a 1, this beats the 11Δ /6 bound of Vigoda for general graphs.

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