Contents

1. Cayley Graphs
2. Vertex-Transitive Graphs
3. Arc-Transitive Graphs
4. Bipartite Graphs and Cycle Parity
5. Tutte's Theorem
6. Tutte's 8-Cage
7. Linear Congruential Generator
8. Numbering Lines

Cayley Graphs

Let \( G \) be a group and let \( C \subseteq G \) such that \( C \) is closed under taking inverses and does not contain the identity, i.e. \[ \forall x \in C, \; x^{-1} \in C, \qquad e \notin C. \] Then the Cayley graph \( X(G, C) \) is the graph with the vertex set \( V(X(G, C)) \) and edge set \( E(X(G, C)) \) defined by \begin{align*} V(X(G, C)) &= G, \\ E(X(G, C)) &= \{ gh : hg^{-1} \in C \}. \end{align*} The set \( C \) is known as the connection set.

Vertex-Transitive Graphs

A graph \( X \) is vertex-transitive if its automorphism group acts transitively on its set of vertices \( V(X). \) Intuitively, this means that no vertex has a special role. We can 'move' the graph around so that any chosen vertex becomes any other vertex. In other words, all vertices are indistinguishable. The graph looks the same from each vertex.

The \( k \)-cube \( Q_k \) is vertex-transitive. So are the Cayley graphs \( X(G, C). \) However the path graph \( P_3 \) is not vertex-transitive since no automorphism can send the middle vertex of valency \( 2 \) to an end vertex of valency \( 1. \)

Arc-Transitive Graphs

The cube \( Q_3 \) is \( 2 \)-arc-transitive but not \( 3 \)-arc-transitive. In \( Q_3, \) a \( 3 \)-arc belonging to a \( 4 \)-cycle cannot be sent to a \( 3 \)-arc that does not belong to a \( 4 \)-cycle. This is easy to explain. The end vertices of a \( 3 \)-arc belonging to a \( 4 \)-cycle are adjacent but the end vertices of a \( 3 \)-arc not belonging to a \( 4 \)-cycle are not adjacent. Therefore, no automorphism can map the end vertices of the first \( 3 \)-arc to those of the second \( 3 \)-arc.

For intuition, imagine that a traveller stands on a vertex and chooses an edge to move along. They do this \( s \) times thereby walking along an arc of length \( s, \) also known as an \( s \)-arc. By the definition of \( s \)-arcs, the traveller is not allowed to backtrack from one vertex to the previous one immediately. In an \( s \)-arc-transitive graph, these arcs look the same no matter which vertex they start from or which edges they choose. In the cube, this is indeed true for \( s = 2. \) All arcs of length \( 2 \) are indistinguishable. No matter which arc of length \( 2 \) the traveller has walked along, the graph would look the same from their perspective at each vertex along the arc. However, this no longer holds good for arcs of length \( 3 \) since there are two distinct kinds of arcs of length \( 3. \) The first kind ends at a distance of \( 1 \) from the starting vertex of the arc (when the arc belongs to a \( 4 \)-cycle). The second kind ends at a distance \( 3 \) from the starting vertex of the arc (when the arc does not belong to a \( 4 \)-cycle). Therefore the cube is not \( 3 \)-arc-transitive.

Bipartite Graphs and Cycle Parity

A graph is bipartite if and only if it contains no cycles of odd length. Equivalently, every cycle in a bipartite graph has even length. Conversely, if every cycle in a graph has even length, then the graph is bipartite.

Tutte's Theorem

For any \( s \)-arc-transitive cubic graph, \( s \le 5. \) This was demonstrated by W. T. Tutte in 1947. A proof can be found in Chapter 18 of Algebraic Graph Theory by Norman Biggs.

In 1973, Richward Weiss established a more general theorem that proves that for any \( s \)-arc-transitive graph, \( s \le 7. \) The bound is weaker but it applies to all graphs rather than only to cubic ones.

Tutte's 8-Cage

The book Algebraic Graph Theory by Godsil and Royle offers the following two descriptions of Tutte's 8-cage on 30 vertices:

Take the cube and an additional vertex \( \infty. \) In each set of four parallel edges, join the midpoint of each pair of opposite edges by an edge, then join the midpoint of the two new edges by an edge, and finally join the midpoint of this edge to \( \infty. \)

Construct a bipartite graph \( T \) with the fifteen edges as one colour class and the fifteen \( 1 \)-factors as the other, where each edge is adjacent to the three \( 1 \)-factors that contain it.

It can be shown that both descriptions construct a cubic bipartite graph on \( 30 \) vertices of girth \( 8. \) It can be further shown that there is a unique cubic bipartite graph on \( 30 \) vertices with girth \( 8. \) As a result both descriptions above construct the same graph.

Linear Congruential Generator

Here is a simple linear congruential generator (LCG) implementation in JavaScript:

function srand (seed) {
  let x = seed
  return function () {
    x = (1664525 * x + 1013904223) % 4294967296
    return x
  }
}

Here is an example usage:

> const rand = srand(0)
undefined
> rand()
1013904223
> rand()
1196435762
> rand()
3519870697

Numbering Lines

Both BSD and GNU cat can number output lines with the -n option. For example:

$ printf 'foo\nbar\nbaz\n' | cat -n
     1  foo
     2  bar
     3  baz

However I have always used nl for this. For example:

$ printf 'foo\nbar\nbaz\n' | nl
     1  foo
     2  bar
     3  baz

While nl is specified in POSIX, the cat -n option is not.

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