quadgraph

A quadgraph is a type of graph in theoretical computer science and mathematics that is defined by its vertices and edges. Specifically, a quadgraph is a graph where each vertex has a degree of exactly four. The degree of a vertex is the number of edges connected to it. Therefore, in a quadgraph, every vertex is incident to four edges.

Quadgraphs are a subset of k-regular graphs, where k is a fixed non-negative integer. In this case, k is equal to 4. The study of regular graphs, including quadgraphs, is important in understanding the structural properties of various networks and combinatorial objects. For instance, in network theory, a quadgraph could represent a network where every node has exactly four connections.

The existence of quadgraphs is not guaranteed for arbitrary numbers of vertices. However, for any even number of vertices greater than or equal to 4, a quadgraph can be constructed. For example, a complete graph on five vertices, denoted K5, is a quadgraph because each of its five vertices is connected to the other four. Another example is the graph formed by the vertices and edges of a tesseract, which is a four-dimensional hypercube. This graph has 16 vertices, and each vertex has a degree of 4, making it a quadgraph. The properties and applications of quadgraphs are explored in various areas of graph theory, including their connectivity, coloring, and embedding on surfaces.

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