Graph Theory: Let S be a set of three pairwise-nonadjacent edges in a 3-connected graph G....

Graph Theory: Let S be a set of three pairwise-nonadjacent edges in a 3-connected graph G. Show that there is a cycle in G containing all three edges of S unless S is an edge-cut of G

Expert Solution

A graph that contains a path between every pair of vertices is connected. Every graph consists of one or more disjoint connected sub-graphs called the connected components. 3-connected graph means that each vertex is connected to two of it's adjacent vertices. An example is shown in image below:

Also, we say that a given graph contains a path (or cycle) of length n if it contains a sub-graph which is isomorphic to Cn , where Cn is is graph of length n. Shown below is and example of such isomorphic sub-graph:

It clearly shows that a cycle exists when all

Similarily for 3-connected graph, G with S as set of three pairwise-nonadjacent edges(AB, CD, EF) is shown in image.

1. and 2. are the isomorphic sub-graphs that are cyclic. But when and edge cut like AD or CE are taken, cycle is not formed.

Hence, proved.

Colby Messinger answered 2 months ago

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