Induced Subgraph in Random Regular Graph

2008 ◽  
Vol 21 (4) ◽  
pp. 645-650
Author(s):  
Lan XIAO ◽  
Guiying YAN ◽  
Yuwen WU ◽  
Wei REN
Keyword(s):  
Regular Graph ◽  
Induced Subgraph ◽  
Random Regular Graph

scholarly journals Optimal Construction of Edge-Disjoint Paths in Random Regular Graphs

2000 ◽  
Vol 9 (3) ◽  
pp. 241-263 ◽  
Author(s):  
ALAN M. FRIEZE ◽  
LEI ZHAO
Keyword(s):  
Polynomial Time ◽  
Regular Graph ◽  
Randomized Algorithm ◽  
Constant Factor ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Edge Disjoint ◽  
Random Regular Graph ◽  
Arbitrary Graphs ◽  
Optimal Construction

Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is [Nscr ][Pscr ]-complete, although it is in [Pscr ] if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n/log n) pairs of vertices can be joined. This is within a constant factor of the optimum.

scholarly journals Expansion properties of a random regular graph after random vertex deletions

2008 ◽  
Vol 29 (5) ◽  
pp. 1139-1150 ◽  
Author(s):  
Catherine Greenhill ◽  
Fred B. Holt ◽  
Nicholas Wormald
Keyword(s):  
Regular Graph ◽  
Random Regular Graph

scholarly journals Dismantling Sparse Random Graphs

2008 ◽  
Vol 17 (2) ◽  
pp. 259-264 ◽  
Author(s):  
SVANTE JANSON ◽  
ANDREW THOMASON
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Regular Graph ◽  
Random Regular Graph

We consider the number of vertices that must be removed from a graphGin order that the remaining subgraph have no component with more thankvertices. Our principal observation is that, ifGis a sparse random graph or a random regular graph onnvertices withn→ ∞, then the number in question is essentially the same for all values ofkthat satisfy bothk→ ∞ andk=o(n).

scholarly journals The analysis of a prioritised probabilistic algorithm to find large induced forests in regular graphs with large girth

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Carlos Hoppen
Keyword(s):  
Lower Bounds ◽  
Regular Graphs ◽  
Probabilistic Algorithms ◽  
Induced Subgraph ◽  
Fixed Integer ◽  
Asymptotic Bounds ◽  
Random Regular Graph ◽  
International Audience ◽  
Large Girth ◽  
Deterministic Bounds

International audience The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.

scholarly journals Induced Forests in Regular Graphs with Large Girth

2008 ◽  
Vol 17 (3) ◽  
pp. 389-410 ◽  
Author(s):  
CARLOS HOPPEN ◽  
NICHOLAS WORMALD
Keyword(s):  
Lower Bound ◽  
Regular Graph ◽  
Randomized Algorithm ◽  
Alternative Proof ◽  
Regular Graphs ◽  
Induced Subgraph ◽  
Large Girth

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomized algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.

scholarly journals Return probability for the Anderson model on the random regular graph

2018 ◽  
Vol 98 (13) ◽  
Author(s):  
Soumya Bera ◽  
Giuseppe De Tomasi ◽  
Ivan M. Khaymovich ◽  
Antonello Scardicchio
Keyword(s):  
Regular Graph ◽  
Anderson Model ◽  
Return Probability ◽  
Random Regular Graph
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