18,656 research outputs found
Continuum Cascade Model of Directed Random Graphs: Traveling Wave Analysis
We study a class of directed random graphs. In these graphs, the interval
[0,x] is the vertex set, and from each y\in [0,x], directed links are drawn to
points in the interval (y,x] which are chosen uniformly with density one. We
analyze the length of the longest directed path starting from the origin. In
the large x limit, we employ traveling wave techniques to extract the
asymptotic behavior of this quantity. We also study the size of a cascade tree
composed of vertices which can be reached via directed paths starting at the
origin.Comment: 12 pages, 2 figures; figure adde
Long induced paths in graphs
We prove that every 3-connected planar graph on vertices contains an
induced path on vertices, which is best possible and improves
the best known lower bound by a multiplicative factor of . We
deduce that any planar graph (or more generally, any graph embeddable on a
fixed surface) with a path on vertices, also contains an induced path on
vertices. We conjecture that for any , there is a
contant such that any -degenerate graph with a path on vertices
also contains an induced path on vertices. We provide
examples showing that this order of magnitude would be best possible (already
for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio
- …