434 research outputs found
Almost spanning subgraphs of random graphs after adversarial edge removal
Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with
p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost
spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth
in the following sense: asymptotically almost surely, if an adversary deletes
arbitrary edges in G(n,p) such that each vertex loses less than half of its
neighbours, then the resulting graph still contains a copy of all such H.Comment: 46 pages, 6 figure
Universality for transversal Hamilton cycles
Let be a graph collection on a common
vertex set of size such that for every
. We show that contains every Hamilton cycle pattern.
That is, for every map there is a Hamilton cycle whose
-th edge lies in .Comment: 18 page
Rainbow Connection of Random Regular Graphs
An edge colored graph is rainbow edge connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connection of
a connected graph , denoted by , is the smallest number of colors
that are needed in order to make rainbow connected.
In this work we study the rainbow connection of the random -regular graph
of order , where is a constant. We prove that with
probability tending to one as goes to infinity the rainbow connection of
satisfies , which is best possible up to a hidden
constant
Darwin's Rainbow: Evolutionary radiation and the spectrum of consciousness
Evolution is littered with paraphyletic convergences: many roads lead to functional Romes. We propose here another example - an equivalence class structure factoring the broad realm of possible realizations of the Baars Global Workspace consciousness model. The construction suggests many different physiological systems can support rapidly shifting, sometimes highly tunable, temporary assemblages of interacting unconscious cognitive modules. The discovery implies various animal taxa exhibiting behaviors we broadly recognize as conscious are, in fact, simply expressing different forms of the same underlying phenomenon. Mathematically, we find much slower, and even multiple simultaneous, versions of the basic structure can operate over very long timescales, a kind of paraconsciousness often ascribed to group phenomena. The variety of possibilities, a veritable rainbow, suggests minds today may be only a small surviving fraction of ancient evolutionary radiations - bush phylogenies of consciousness and paraconsciousness. Under this scenario, the resulting diversity was subsequently pruned by selection and chance extinction. Though few traces of the radiation may be found in the direct fossil record, exaptations and vestiges are scattered across the living mind. Humans, for instance, display an uncommonly profound synergism between individual consciousness and their embedding cultural heritages, enabling efficient Lamarkian adaptation
Rainbow subgraphs of uniformly coloured randomly perturbed graphs
For a given , the randomly perturbed graph model is defined
as the union of any -vertex graph with minimum degree and
the binomial random graph on the same vertex set. Moreover,
we say that a graph is uniformly coloured with colours in if each
edge is coloured independently and uniformly at random with a colour from
.
Based on a coupling idea of McDiarmird, we provide a general tool to tackle
problems concerning finding a rainbow copy of a graph in a uniformly
coloured perturbed -vertex graph with colours in . For
example, our machinery easily allows to recover a result of Aigner-Horev and
Hefetz concerning rainbow Hamilton cycles, and to improve a result of
Aigner-Horev, Hefetz and Lahiri concerning rainbow bounded-degree spanning
trees.
Furthermore, using different methods, we prove that for any and integer , there exists such that the
following holds. Let be a tree on vertices with maximum degree at most
and be an -vertex graph with . Then a
uniformly coloured with colours in
contains a rainbow copy of with high probability. This is optimal both in
terms of colours and edge probability (up to a constant factor).Comment: 22 pages, 1 figur
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