Immense Subcubic Graphic Numbers - Numberphile

Key Concepts

• Simple Graph: Nodes (vertices) connected by edges, without loops (node connected to itself) or parallel edges (multiple edges between two nodes).
• Subcubic Graph: A simple graph where each node has a degree of at most 3 (0, 1, 2, or 3 edges connected to it).
• Hiding (Topological Minor): One graph is "hiding" inside another if a copy of the first graph can be found within the second, including cases where edges are subdivided by adding nodes.
• Subcubic Graph Number (SSCGn): The maximum length of a sequence of subcubic simple graphs where the i-th graph has at most n+i-1 nodes, and no graph in the sequence is hiding in any later graph.
• General Subcubic Graph: Subcubic graphs that allow loops and parallel edges.
• Graph Minor Theorem: In any infinite collection of finite graphs, there will always be two graphs, A and B, where A is a minor of B. A is a minor of B if A can be obtained from B by deleting edges, deleting nodes, and contracting edges.
• Edge Contraction: An operation where an edge is "squeezed" away, merging the two vertices it connected into a single vertex.
• TREE(n): A sequence of extremely fast-growing numbers in combinatorics. TREE(3) is already unimaginably large.

Subcubic Graph Numbers

Definition and Initial Examples

The video introduces the concept of subcubic graph numbers. To understand these numbers, the speaker first defines a simple graph as a collection of nodes (vertices) connected by edges, with the restriction that there are no loops (an edge connecting a node to itself) and no parallel edges (more than one edge connecting the same two nodes). A subcubic graph is then defined as a simple graph where each node has at most three edges connected to it.

The concept of one graph "hiding" inside another is explained. This means that a copy of the first graph can be found within the second graph, even if edges in the first graph are subdivided by adding nodes along them. The technical term for this is being a topological minor.

The subcubic graph number, denoted as SSCGn, represents the maximum length of a sequence of subcubic simple graphs with specific constraints. The i-th graph in the sequence can have at most n+i-1 nodes, and no graph in the sequence can be "hiding" in any of the subsequent graphs.

The video then explores some initial examples:

• SSCG0: Starting with at most one node, the longest sequence is two: a single node graph, followed by the empty graph (no nodes or edges). Any graph with nodes will have the single node graph "hiding" inside it, and any graph with edges will have the empty graph "hiding" inside it.
• SSCG1: Starting with at most two nodes, the longest sequence is five: a graph with two nodes connected by an edge, followed by a graph with three nodes and no edges, then two nodes and no edges, one node and no edges, and finally the empty graph.

SSCG2 and the Work of Adam Goucher

The video then considers SSCG2, where the first graph in the sequence can have at most three nodes. The speaker suggests starting with a triangle graph (three nodes connected to form a triangle). Brady guesses that the length of the sequence will be 27. However, the actual value is much larger, estimated to be greater than 10^(10^28) by Adam Goucher.

Adam Goucher's work is highlighted. He constructed a sequence of this length, but it's not proven to be the absolute longest possible sequence. The video emphasizes that these sequences will always end due to deeper mathematical principles.

SSCG3 and Beyond

The video moves on to SSCG3, where the first graph can have at most four nodes. The speaker suggests starting with a complete graph with four nodes. The size of SSCG3 is described as "very, very big," even larger than the famously huge number TREE(3).

Harvey Friedman is mentioned as the logician who invented these sequences and has generated some of the largest numbers in mathematics.

General Subcubic Graph Numbers

The video then explores a variation where the graphs are no longer required to be simple. This means loops and parallel edges are allowed. This change leads to even larger subcubic graph numbers.

• SSCG0 (General): The longest sequence is now six. The sequence can include a single node with a loop, followed by two nodes connected by a single edge, then three nodes and no edges, two nodes and no edges, one node and no edges, and finally the empty graph.
• SSCG1 (General): The value is significantly larger than the simple graph case (which was 5), already entering the realm of enormous numbers, exceeding Graham's number.
• SSCG3 (General): This number is much larger than TREE(3), exceeding even TREE(TREE(3)). It's described as TREE^3(3), where the tree function is applied three times.

The speaker emphasizes that these numbers are so large that it's difficult to comprehend their scale, even with powerful notations like Knuth's up-arrow notation.

The Graph Minor Theorem and its Significance

The video transitions from exploring the Google of large numbers to discussing the deeper mathematical significance behind these numbers. The key result is the Graph Minor Theorem, proven by Neil Robertson and Paul Seymour in a series of 20 papers.

The Graph Minor Theorem states that in any infinite collection of finite graphs, there will always be two graphs, A and B, where A is a minor of B. A is a minor of B if A can be obtained from B by deleting edges, deleting nodes, and contracting edges. Edge contraction involves "squeezing" an edge away, merging the two vertices it connected.

The video notes that the notion of "hiding" used earlier is equivalent to being a minor for subcubic graphs.

The Graph Minor Theorem is described as a profound result in graph theory, dwarfing other results in the field. The appearance of extremely large numbers in the context of subcubic graph numbers is linked to the deep mathematical principles underlying the Graph Minor Theorem.

Conclusion

The video explores the fascinating world of subcubic graph numbers, demonstrating how seemingly simple rules can lead to unimaginably large numbers. It connects these numbers to the Graph Minor Theorem, highlighting the deep mathematical principles at play. The video emphasizes that the study of these numbers is not just about playing with large quantities but also about uncovering fundamental truths about the structure of graphs.