Question

(i) Let \(G\) be a graph with four vertices and degree sequence \((1,2,3,4)\). Write down the number of edges of \(G\), and construct such a graph. (ii) Are there any simple graphs with four vertices and degree sequence \((1,2,3,4) ?\)

Short answer

The graph \(G\) has 5 edges. It is possible to construct such a graph, but it's impossible to construct a simple graph with four vertices and degree sequence \((1,2,3,4)\).

Step-by-step solution

Definition reminders for the Graph & Degree Sequence

A Graph is a set of vertices and a collection of edges that each connects a pair of vertices. Degree Sequence is a list of the degrees of the vertices in the graph in non-increasing order.

Calculate the number of edges

The sum of the degree sequences is twice the number of edges (by Handshaking Lemma). Therefore, adding the degree sequence, \(1+2+3+4 = 10\), and dividing by two gives the number of edges. So, we have \(10/2 = 5\) edges in the graph.

Construct the graph

Draw 4 vertices. Connect the vertices such that one vertex has 1 edge (degree 1), one vertex has 2 edges (degree 2), one has 3 edges (degree 3), and one has 4 edges (degree 4). Note that vertex with 4 edges is connected to all the vertices.

Analysis for simple graphs

A simple graph is one with no self-loops and no more than one edge between any two vertices. Looking at the degree sequence, we can observe that one of the vertices has degree 4. But since there are only 4 vertices in the graph, and no vertex can be connected to itself, the maximum degree any vertex can have in a simple graph of 4 vertices is 3. Therefore, there are no simple graphs with four vertices and degree sequence of \((1,2,3,4)\).

Key concepts

Degree Sequence in Graphs

In graph theory, a degree sequence is a list of degrees of the vertices in a graph, typically written in non-increasing order. Each degree indicates the number of edges connected to a vertex. For example, in a degree sequence of \((1,2,3,4)\), there is one vertex with degree 1, one with degree 2, one with degree 3, and one with degree 4.
Understanding degree sequences is crucial as it provides insight into the structure and potential configurations of a graph. The sum of the degrees in the sequence should always be even because each edge contributes to the degree of two vertices. This fact is highlighted by the Handshaking Lemma, which states that the total sum of all vertex degrees is twice the number of edges in a graph. Hence, for checking if a graph fits a degree sequence, ensure the sum is valid for graph formation. This sequence particularly challenges when constructing a simple graph, where certain arrangements like self-loops are disallowed.

Graph Construction

Constructing a graph involves creating a set of vertices and connecting them with edges as specified in the degree sequence. Begin by placing the number of vertices required. For the degree sequence \((1,2,3,4)\), four vertices are positioned initially.
Next, you assign connections (edges) between these vertices. Follow the degree needs: one vertex connects to 1 other vertex, another connects to 2 others, one to 3, and finally, one to all 4 vertices. It requires adjusting and checking pairings to satisfy all conditions of the sequence.
The main challenge is respecting degrees without breaking graph rules, like ensuring no vertex repeats a connection already established. This is feasible in a general graph situation, but as you'll see, limitations arise with simple graphs. Creative solutions might involve considering indirect or weighted graphs that differ from simple, direct connections.

Understanding Simple Graphs

Simple graphs are a fundamental concept in graph theory. They are defined by their structure, where each edge connects two distinct vertices, with no loops (edges that start and end on the same vertex), and no multiple edges between the same pair of vertices. This can limit certain degree sequences from forming a valid simple graph.
The key point discussed for \((1,2,3,4)\) is a vertex degree limitation. In a graph of four vertices, the highest degree any vertex could have is three because no vertex can connect to itself. Since simple graphs do not permit more than one edge between any vertices, these constraints make a sequence like \((1,2,3,4)\) impossible without violating simple graph rules.
Familiarizing with these rules helps in problem-solving, illustrating when complications arise in graph construction, and expanding your understanding by considering modifications or alternative sequences or graph types.