Let (d1, d2, …, dn) be a sequence of n nonnegative integers...

Let $\left(d_{1}, d_{2}, \ldots, d_{n}\right)$ be a sequence of $n$ nonnegative integers whose sum $d_{1}+d_{2}+$ $\cdots+d_{n}$ is even. Prove that there exists a general graph with this sequence as its degree sequence. Devise an algorithm to construct such a general graph.

Key Concepts

Graph Realization Problem

The graph realization problem asks whether there exists a graph that corresponds to a given degree sequence. It is a classical problem in graph theory that leads to deeper insights into the structure and properties of graphs. The problem is central when one is given only the degrees of the vertices and must construct a graph that meets these constraints, which often requires both theoretical criteria and constructive algorithms.

Greedy Graph Construction

Greedy graph construction refers to the iterative and often algorithmic approach of connecting vertices based on their remaining degree requirements. In the context of realizing a degree sequence, a greedy strategy typically involves repeatedly selecting vertices with higher degree demands and connecting them to satisfy their edge requirements. This approach not only provides a constructive proof of the sequence's realizability but also lends itself to algorithmic implementation.

Degree Sequence

A degree sequence is a list of nonnegative integers that represent the degrees of the vertices in a graph. In graph theory, understanding whether a given sequence can be realized as the degree sequence of some graph is a fundamental question. This involves ensuring that the distribution of degrees is possible under the constraints of how edges can connect vertices, especially taking into account the structure of the graph (for instance, simple graphs have no loops or multiple edges).

Handshake Lemma

The handshake lemma states that the sum of the degrees of all vertices in any graph is equal to twice the number of edges, and thus must be even. This condition is necessary for a sequence of degrees to be graphical (i.e., to be the degree sequence of some graph). This lemma provides the first quick check when determining if any sequence is eligible to be a degree sequence.

Havel-Hakimi Algorithm

The Havel-Hakimi algorithm is a well-known constructive procedure used in graph theory to determine if a degree sequence is graphical and to construct a corresponding graph if it is. The algorithm works by repeatedly removing the vertex with the highest degree and reducing the degrees of the next highest vertices accordingly. This iterative process ensures that if the sequence can be realized as a graph, the algorithm will provide a method to build it.

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