Nov30-10, 08:21 PM
Problem 4. Let k be an integer and let D be a directed graph with the property that
deg+(v) = k = deg-(v) for every v IN V (D). Prove that there exist vertex disjoint directed
cycles C1,...Ck so that SUm of V(Ci) = V (D). (HInt: construct a bipartite graph H from D
so that each vertex in D splits into two vertices in H.)
No idea how to even start...
Def: A transitive tournament is a tournament with no directed cycles. Equivalently, it is
a tournament so that the vertices can be ordered v1; v2,.....vn so that (vi, vj) is an edge
whenever i < j.
Problem 6. Let T be a tournament on n vertices. Prove that T contains a subgraph which is a transitive tournament on log2n + 1 vertices.
I have no idea how to do it, i made a ton of graphs with up to 8 vertices and found subgraphs that work, but can't find any way to prove this.
|Register to reply|
|Converting Velocity-Time Graphs Into Acceleration Graphs||Introductory Physics Homework||5|
|Good software for making directed graphs?||Academic Guidance||3|
|Very simple calculus problem...graphs and velocity/time graphs to acceleration.||Calculus & Beyond Homework||1|
|directed sets||Calculus & Beyond Homework||3|
|Equilibrium Points of Directed Graphs||General Math||4|