29.1.9. Question 8

A simple graph is called regular if the degree of every vertex is the same. And if the degree of every vertex in a simple graph is some value r, then we call such a graph an r-regular graph.

A regular graph is defined based on the degrees of its vertices. The degree of a vertex is the number of edges connected to it. In a regular graph, every vertex has the same degree, denoted as r. For example, in a 3-regular graph, every vertex is connected to exactly three edges.

Imagine a classroom with students (vertices) where each student knows exactly three other students (the edges). This scenario forms a 3-regular graph. If each student knew only two others, it would form a 2-regular graph.

Here, you are given a value of k. You have to draw a simple regular graph where the degree of every vertex is 2 times k + 1 such that the graph has a cut edge.

The task involves creating a k-regular graph where each vertex has a degree of 2k + 1. This means each vertex is connected to 2k + 1 other vertices. The construction ensures that some edges act as cut edges, meaning if one of them is removed, the graph becomes disconnected.

Think of a social group where every person has 2k friends, plus 1 acquaintance outside their primary social circle. If you remove the acquaintance, it may cut off connections to a part of the group, thus 'cutting' the friendship network into parts.

To construct the graph, we take two copies of a complete bipartite graph where I have 2k nodes in the individual sets in the bipartition.

We start by creating two copies of a complete bipartite graph. In each bipartite graph, we have two sets of nodes, each containing 2k nodes. Each node from one set connects to every node in the other set. This guarantees a dense interconnectedness within the individual sets.

Imagine two teams of players. Each player from Team A (2k players) knows or interacts with every player from Team B (another set of 2k players). This setup makes sure that everyone is well connected, representing our bipartite graph.

To introduce a cut edge, I connect one new vertex to all vertices in one set of the bipartite graph.

A cut edge is added by linking a new vertex to all members of one set in the bipartite graph. This new vertex will have a degree of 2k + 1, as it connects to all 2k vertices in that set plus one extra edge to another vertex in the adjacent set.

Picture adding a new team leader who is involved with every team member but also has a connection to a member of another team. If the leader is removed, communication between these teams may break, signifying a cut edge.

We ensure that every vertex in both sets has the degree of 2k + 1 by pairing them and adding edges between pairs.

To finalize the regularity of the graph, additional edges are added between pairs of nodes within each set. This ensures each node reaches the required degree, confirming every vertex meets the condition of being a regular graph.

Continuing with our team analogy, suppose after adding connections between all players, you also ensure that every player within each team knows two more teammates. This way, everyone has equal connections, fulfilling the regularity condition.

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