Arboricity and spanning-tree packing in random graphs

2017 ◽  
Vol 52 (3) ◽  
pp. 495-535 ◽  
Author(s):  
Pu Gao ◽  
Xavier Pérez-Giménez ◽  
Cristiane M. Sato
Keyword(s):  
Random Graphs ◽  
Spanning Tree ◽  
Tree Packing

scholarly journals The Size of the Giant Joint Component in a Binomial Random Double Graph

10.37236/8846 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Mark Jerrum ◽  
Tamás Makai
Keyword(s):  
Phase Transition ◽  
Critical Point ◽  
Random Graphs ◽  
Spanning Tree ◽  
Branching Process ◽  
Graph Model ◽  
First Order ◽  
Blue Edge ◽  
Critical Pairs

We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices.  A joint component is a maximal set of vertices that supports both a red and a blue spanning tree.  We show that there are critical pairs of red and blue edge densities at which a giant joint component appears.  In contrast to the standard binomial graph model, the phase transition is first order:  the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point.  We connect this phenomenon to the properties of a certain bicoloured branching process. 

scholarly journals Fractional spanning tree packing, forest covering and eigenvalues

2016 ◽  
Vol 213 ◽  
pp. 219-223 ◽  
Author(s):  
Yanmei Hong ◽  
Xiaofeng Gu ◽  
Hong-Jian Lai ◽  
Qinghai Liu
Keyword(s):  
Spanning Tree ◽  
Tree Packing

Critical random graphs and the structure of a minimum spanning tree

2008 ◽  
Vol 35 (3) ◽  
pp. 323-347 ◽  
Author(s):  
L. Addario-Berry ◽  
N. Broutin ◽  
B. Reed
Keyword(s):  
Random Graphs ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Critical Random Graphs

scholarly journals Power of $k$ Choices and Rainbow Spanning Trees in Random Graphs

10.37236/4642 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Alan Frieze ◽  
Paweł Prałat
Keyword(s):  
Stochastic Process ◽  
Random Graph ◽  
Random Graphs ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Special Importance ◽  
Sharp Threshold

We consider the Erdős-Rényi random graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new edge chosen uniformly at random from the set of missing edges. Let $\mathcal{G}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\ldots, m$) of $\mathcal{G}(n,m)$ independently chooses precisely $k \in\mathbb{N}$ colours, uniformly at random, from a given set of $n-1$ colours (one may view $e_i$ as a multi-edge). We stop the process prematurely at time $M$ when the following two events hold: $\mathcal{G}(n,M)$ is connected and every colour occurs at least once ($M={n \choose 2}$ if some colour does not occur before all edges are present; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\mathcal{G}(n,M)$ has a rainbow spanning tree (that is, multicoloured tree on $n$ vertices). Clearly, both properties are necessary for the desired tree to exist.In 1994, Frieze and McKay investigated the case $k=1$ and the answer to this question is "yes" (asymptotically almost surely). However, since the sharp threshold for connectivity is $\frac {n}{2} \log n$ and the sharp threshold for seeing all the colours is $\frac{n}{k} \log n$, the case $k=2$ is of special importance as in this case the two processes keep up with one another. In this paper, we show that asymptotically almost surely the answer is "yes" also for $k \ge 2$.

Spanning tree packing number and eigenvalues of graphs with given girth

2019 ◽  
Vol 578 ◽  
pp. 411-424 ◽  
Author(s):  
Ruifang Liu ◽  
Hong-Jian Lai ◽  
Yingzhi Tian
Keyword(s):  
Spanning Tree ◽  
Packing Number ◽  
Tree Packing ◽  
Eigenvalues Of Graphs

scholarly journals Note on the spanning-tree packing number of lexicographic product graphs

2015 ◽  
Vol 338 (5) ◽  
pp. 669-673 ◽  
Author(s):  
Hengzhe Li ◽  
Xueliang Li ◽  
Yaping Mao ◽  
Jun Yue
Keyword(s):  
Spanning Tree ◽  
Lexicographic Product ◽  
Packing Number ◽  
Product Graphs ◽  
Tree Packing ◽  
Lexicographic Product Graphs

Random Graphs

1998 ◽  
Author(s):  
V. F. Kolchin
Keyword(s):  
Random Graphs

Self-Protected Spanning Tree Based Recovery Scheme to Protect against Single Failure

2009 ◽  
Vol E92-B (3) ◽  
pp. 909-921
Author(s):  
Depeng JIN ◽  
Wentao CHEN ◽  
Li SU ◽  
Yong LI ◽  
Lieguang ZENG
Keyword(s):  
Spanning Tree ◽  
Recovery Scheme
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