11 Graph Properties and Applications Free Online FlashCards

What does an Eulerian path use exactly once?

An Eulerian path uses every edge exactly once. An Eulerian circuit is an Eulerian path that begins and ends at the same vertex.

When does a connected graph have an Eulerian circuit?

A connected undirected graph has an Eulerian circuit if and only if every vertex has even degree.

What degree pattern gives an open Eulerian path?

A connected undirected graph has an Eulerian path but no circuit exactly when two vertices have odd degree; the path starts and ends at those vertices.

What is a Hamiltonian path?

A Hamiltonian path visits every vertex exactly once. Unlike an Eulerian path, it is concerned with vertices rather than edges.

What degree condition is necessary for a Hamiltonian cycle?

Every vertex in a Hamiltonian cycle must have degree at least 2, because the cycle enters and leaves each vertex.

What graph structure represents a feasible TSP tour?

A feasible TSP tour is a Hamiltonian cycle, and the objective is to minimize its total edge weight.

What is Euler’s formula for a connected plane graph?

For a connected plane graph, Euler’s formula is $|V|-|E|+|F|=2$, where $|F|$ includes the unbounded exterior face.

What makes a graph planar?

A planar graph can be drawn in the plane with no edge crossings except at shared endpoints. The property belongs to the abstract graph, not one particular drawing.

Which two graphs characterize nonplanarity in Kuratowski’s theorem?

Kuratowski’s theorem states that a graph is planar exactly when it contains no subdivision of $K_5$ or $K_{3,3}$.

What is the chromatic number?

A proper vertex coloring assigns different colors to adjacent vertices. The chromatic number $\chi(G)$ is the minimum number of colors needed.

How does vertex coloring model exam scheduling?

In exam scheduling, exams are vertices and edges connect exams sharing students. A proper coloring assigns time slots so adjacent exams do not occur simultaneously.

How do maximal, maximum, and perfect matchings differ?

A matching is maximal if no edge can be added, maximum if it has the greatest possible size, and perfect if every vertex belongs to exactly one selected edge.