Edge percolation on a random regular graph of low degree

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.

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Expansion properties of a random regular graph after random vertex deletions

European Journal of Combinatorics ◽

10.1016/j.ejc.2007.06.021 ◽

2008 ◽

Vol 29 (5) ◽

pp. 1139-1150 ◽

Cited By ~ 4

Author(s):

Catherine Greenhill ◽

Fred B. Holt ◽

Nicholas Wormald

Keyword(s):

Regular Graph ◽

Random Regular Graph

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On the Dynamics of Branching Random Walk on Random Regular Graph

2020 International Conference on Information Networking (ICOIN) ◽

10.1109/icoin48656.2020.9016620 ◽

2020 ◽

Author(s):

Takuro Mayaguchi ◽

Ryota Sakaguchi ◽

Hiroyuki Ohsaki

Keyword(s):

Random Walk ◽

Regular Graph ◽

Branching Random Walk ◽

Random Regular Graph

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Evolution of Cooperation for Multiple Mutant Configurations on All Regular Graphs with N ≤ 14 Players

Games ◽

10.3390/g11010012 ◽

2020 ◽

Vol 11 (1) ◽

pp. 12

Author(s):

Hendrik Richter

Keyword(s):

Regular Graph ◽

Good Predictor ◽

Computational Approach ◽

Structured Populations ◽

Evolution Of Cooperation ◽

Regular Graphs ◽

Weak Selection ◽

Structure Coefficient ◽

Low Degree ◽

Emergence Of Cooperation

We study the emergence of cooperation in structured populations with any arrangement of cooperators and defectors on the evolutionary graph. In a computational approach using structure coefficients defined for configurations describing such arrangements of any number of mutants, we provide results for weak selection to favor cooperation over defection on any regular graph with N ≤ 14 vertices. Furthermore, the properties of graphs that particularly promote cooperation are analyzed. It is shown that the number of graph cycles of a certain length is a good predictor for the values of the structure coefficient, and thus a tendency to favor cooperation. Another property of particularly cooperation-promoting regular graphs with a low degree is that they are structured to have blocks with clusters of mutants that are connected by cut vertices and/or hinge vertices.

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Induced Subgraph in Random Regular Graph

Journal of Systems Science and Complexity ◽

10.1007/s11424-008-9141-7 ◽

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

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Dismantling Sparse Random Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548307008802 ◽

2008 ◽

Vol 17 (2) ◽

pp. 259-264 ◽

Cited By ~ 8

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).

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