5.4: Paths
Paths
A graph is a walk if and only if the vertices can be labeled \(v_0, v_1, \ldots, v_k\) such that \(v_i, v_{i+1}\) is an edge.
A graph is a trail if and only if it is a walk such that no edge is used twice.
A graph is a path if and only if it is a walk such that no vertex is used twice.
A path on \(n\) vertices is denoted \(P_n.\) Please note that these terms ( walk , trail , and path ) vary widely including reversing the names, so always check the definition in the article, book, or other material you are reading.
A trail is Eulerian if and only if it uses every edge in the graph.
A path is Hamiltonian if and only if it uses every vertex in that graph.
Practice
Checkpoint \(\PageIndex{6}\)
Draw a path of length 5. Note this is denoted \(P_5.\)
Checkpoint \(\PageIndex{8}\)
Find a walk that is not a trail in Figure \(\PageIndex{7}\)
Checkpoint \(\PageIndex{9}\)
Find a trail that is not a path in Figure \(\PageIndex{7}\)
Checkpoint \(\PageIndex{10}\)
Determine which graphs in Figure 5.2.43 have Eulerian trails.
Checkpoint \(\PageIndex{11}\)
Determine which graphs in Figure 5.2.43 have Hamiltonian paths.
Connected
A graph \(G\) is connected if and only if for every pair of vertices \(v,w\) there exists a path from \(v\) to \(w.\)
A graph \(G\) is \(n\)-connected if and only if removing any \(n-1\) vertices does not disconnect the graph.
Practice
Checkpoint \(\PageIndex{14}\)
Determine which of the graphs in Figure 5.1.1 are connected.
Checkpoint \(\PageIndex{15}\)
Explain why every complete graph is connected.
Checkpoint \(\PageIndex{16}\)
Explain why every complete bipartite graph is connected.
Checkpoint \(\PageIndex{17}\)
Determine if every bipartite graph must be connected.
Checkpoint \(\PageIndex{18}\)
For each of the graphs in Figure 5.2.43 and Figure 5.2.44 determine the maximum \(n\) for which the graph is \(n\)-connected.
Checkpoint \(\PageIndex{19}\)
If a graph is \(n\)-connected what does this say about the minimum of the number of paths between any two vertices?