A 4-dimensional graph has at least 9 edges

A 4-dimensional graph has at least 9 edges

The open problem posed by Paul Erd®s asking for the smallest number of edges in a 4-dimensional graph is solved by showing that a 4-dimensional graph must have at least 9 edges. Furthermore, there is only one 4-dimensional graph with 9 edges, namely K 3,3 .

The Maximum Number of Edges in a Strongly Multiplicative Graph

Removable edges in a cycle of a 4-connected graph

Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G − e; second, for all the vertices x of degree 3 in G− e, delete x from G− e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G a e. If G a...

The number of removable edges in a 4-connected graph

Let G be a 4-connected graph. For an edge e of G; we do the following operations on G: first, delete the edge e from G; resulting the graph G e; second, for all the vertices x of degree 3 in G e; delete x from G e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G~e: If G~e is st...

A Family of Number Fields with Unit Rank at Least 4 That Has Euclidean Ideals

We will prove that if the unit rank of a number field with cyclic class group is large enough and if the Galois group of its Hilbert class field over Q is abelian, then every generator of its class group is a Euclidean ideal class. We use this to prove the existence of a non-principal Euclidean ideal class that is not norm-Euclidean by showing that Q( √ 5, √ 21, √ 22) has such an ideal class.