10 research outputs found
On random k-out sub-graphs of large graphs
We consider random sub-graphs of a fixed graph with large minimum
degree. We fix a positive integer and let be the random sub-graph
where each independently chooses random neighbors, making
edges in all. When the minimum degree then is -connected w.h.p. for ;
Hamiltonian for sufficiently large. When , then has
a cycle of length for . By w.h.p. we mean
that the probability of non-occurrence can be bounded by a function
(or ) where
Crux and long cycles in graphs
We introduce a notion of the \emph{crux} of a graph , measuring the order
of a smallest dense subgraph in . This simple-looking notion leads to some
generalisations of known results about cycles, offering an interesting paradigm
of `replacing average degree by crux'. In particular, we prove that
\emph{every} graph contains a cycle of length linear in its crux.
Long proved that every subgraph of a hypercube (resp. discrete torus
) with average degree contains a path of length (resp.
), and conjectured that there should be a path of length
(resp. ). As a corollary of our result, together with isoperimetric
inequalities, we close these exponential gaps giving asymptotically optimal
bounds on long paths in hypercubes, discrete tori, and more generally Hamming
graphs.
We also consider random subgraphs of -free graphs and hypercubes,
proving near optimal bounds on lengths of long cycles
Long paths and cycles in random subgraphs of graphs with large minimum degree
For a given finite graph of minimum degree at least , let be a
random subgraph of obtained by taking each edge independently with
probability . We prove that (i) if for a function
that tends to infinity as does, then
asymptotically almost surely contains a cycle (and thus a path) of length at
least , and (ii) if , then
asymptotically almost surely contains a path of length at least . Our
theorems extend classical results on paths and cycles in the binomial random
graph, obtained by taking to be the complete graph on vertices.Comment: 26 page