Welcome to our exploration of a master level question in discrete mathematics, dissecting its intricacies and unveiling its solutions. As a discrete math assignment solver, we delve into the theoretical depths of this fascinating field, offering clarity and expertise to unravel complex problems.

Question:

Consider a directed graph $G=(V,E)$ with $n$ vertices and $m$ edges. Prove that if every vertex in $G$ has outdegree at least $n/2$, then $G$ contains a directed cycle.

Answer:

To demonstrate this proposition, we'll employ a proof by contradiction. Suppose $G$ is a directed graph where every vertex has an outdegree of at least $n/2$, yet $G$ does not contain a directed cycle.

Since $G$ is acyclic, it must be a directed acyclic graph (DAG). Let's consider the vertex with the minimum outdegree, denoted as $v$. By our assumption, $v$ has at least $n/2$ outgoing edges.

Now, let's explore the vertices reachable from $v$ via a directed path. Since $G$ is acyclic, this directed path forms a linear chain of vertices. Let's denote the endpoint of this chain as $w$. Since $v$ has at least $n/2$ outgoing edges, and all edges lead to distinct vertices (as $G$ is acyclic), $w$ must have at least $n/2$ incoming edges.

Consider the set of vertices reachable from $w$ via directed paths. Since $G$ is acyclic, this set forms a directed subtree rooted at $w$. Let's denote this subtree as $T$. Since $w$ has at least $n/2$ incoming edges, and each edge leads to a distinct vertex (as $G$ is acyclic), $T$ must contain at least $n/2$ vertices other than $w$.

However, this implies that the subtree $T$ contains more vertices than there are in the entire graph $G$, which is a contradiction. Thus, our initial assumption that $G$ does not contain a directed cycle must be false.

Therefore, we conclude that if every vertex in a directed graph $G$ has an outdegree of at least $n/2$, then $G$ must contain a directed cycle.

Conclusion:

In mastering discrete mathematics assignments, it's crucial to grasp the theoretical underpinnings of complex problems. By unraveling the intricacies of directed graphs and cycles, we gain insights into fundamental concepts that underpin the discipline. As discrete math assignment solvers, we navigate these theoretical landscapes with precision and expertise, illuminating the path to understanding for students and enthusiasts alike.