30,815 research outputs found
Random triangle removal
Starting from a complete graph on vertices, repeatedly delete the edges
of a uniformly chosen triangle. This stochastic process terminates once it
arrives at a triangle-free graph, and the fundamental question is to estimate
the final number of edges (equivalently, the time it takes the process to
finish, or how many edge-disjoint triangles are packed via the random greedy
algorithm). Bollob\'as and Erd\H{o}s (1990) conjectured that the expected final
number of edges has order , motivated by the study of the Ramsey
number . An upper bound of was shown by Spencer (1995) and
independently by R\"odl and Thoma (1996). Several bounds were given for
variants and generalizations (e.g., Alon, Kim and Spencer (1997) and Wormald
(1999)), while the best known upper bound for the original question of
Bollob\'as and Erd\H{o}s was due to Grable (1997). No nontrivial
lower bound was available.
Here we prove that with high probability the final number of edges in random
triangle removal is equal to , thus confirming the 3/2 exponent
conjectured by Bollob\'as and Erd\H{o}s and matching the predictions of Spencer
et al. For the upper bound, for any fixed we construct a family of
graphs by gluing triangles sequentially
in a prescribed manner, and dynamically track all homomorphisms from them,
rooted at any two vertices, up to the point where edges
remain. A system of martingales establishes concentration for these random
variables around their analogous means in a random graph with corresponding
edge density, and a key role is played by the self-correcting nature of the
process. The lower bound builds on the estimates at that very point to show
that the process will typically terminate with at least edges
left.Comment: 42 pages, 4 figures. Supercedes arXiv:1108.178
Optimal subgraph structures in scale-free configuration models
Subgraphs reveal information about the geometry and functionalities of
complex networks. For scale-free networks with unbounded degree fluctuations,
we obtain the asymptotics of the number of times a small connected graph
occurs as a subgraph or as an induced subgraph. We obtain these results by
analyzing the configuration model with degree exponent and
introducing a novel class of optimization problems. For any given subgraph, the
unique optimizer describes the degrees of the vertices that together span the
subgraph.
We find that subgraphs typically occur between vertices with specific degree
ranges. In this way, we can count and characterize {\it all} subgraphs. We
refrain from double counting in the case of multi-edges, essentially counting
the subgraphs in the {\it erased} configuration model.Comment: 50 pages, 2 figure
Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques
For graphs and , we say is Ramsey for if every -coloring of
the edges of contains a monochromatic copy of . The graph is Ramsey
-minimal if is Ramsey for and there is no proper subgraph of
so that is Ramsey for . Burr, Erdos, and Lovasz defined to
be the minimum degree of over all Ramsey -minimal graphs . Define
to be a graph on vertices consisting of a complete graph on
vertices and one additional vertex of degree . We show that
for all values ; it was previously known that , so it
is surprising that is much smaller.
We also make some further progress on some sparser graphs. Fox and Lin
observed that for all graphs , where is
the minimum degree of ; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs without
isolated vertices satisfy . Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that -regular -connected
triangle-free graphs , with one extra technical constraint, satisfy ; the extra constraint is that has a vertex so that if one
removes and its neighborhood from , the remainder is connected.Comment: 10 pages; 3 figure
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