Meredith Graph

[Martez Fields]

TheMeredith graph is 4-vertex-connected and 4-edge-connected, has chromatic number 3, chromatic index 5, radius 7, diameter 8, girth 4 and is non-Hamiltonian.[2] It has book thickness 3 and queue number 2.[3]

Published in 1973, it provides a counterexample to the Crispin Nash-Williams conjecture that every 4-regular 4-vertex-connected graph is Hamiltonian.[4][5] However, W. T. Tutte showed that all 4-connected planar graphs are hamiltonian.[6]

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The answer to the first question is no. To see this, we only need to prove that the constructions by Meredith in [1] give $k$-edge-connected graphs, as Meredith already proved that the graph is non-Hamiltonian and regular.

Lovasz conjectured that every finite connected vertex-transitive graph contains a Hamiltonian cycle except five known counterexamples, i.e. $K_2$, the Petersen graph, the Coxeter graph, and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle. All of them have vertex degree $\leq 3$. As the conjecture is neither proven or refuted, we can say that the answer to the second question is unknown.

We could also put the flour after the eggs, since the only thing that depends on eggs is cake. Since we can rearrange the items, we could also fulfill some of them in parallel while maintaining that no item appears before anything that depends on it. For example, by adding a level of nesting, we can indicate that everything within an inner slice is independent of anything else in that slice:

We also need to be able to remove a node and all edges from a graph. Removing the node is simple, as is removing the outbound edges from each node. However, the fact that we keep track of every edge in both directions means that we have to do a little extra work to remove the inbound records. The strategy that we will use to remove all edges is as follows:

All of that work was not exactly a piece of cake, but now we have a dependency graph that can be used to topologically sort just about anything. You can find the full code for this post on GitHub. There are some notable limitations to this implementation, and I would like to challenge you to improve it so that it can:

Thought leadership with a focus on providing actionable information for investors and managers;

Industry leading research on emerging managers and diversity investing;

Hedge fund due-diligence and ongoing monitoring;

Hedge fund, private equity venture capital and FOFs marketing and PR;

Index construction;

Hedge fund data, investment analysis and investment software;

Asset allocation, and;

Public speaking, writing and conference content development

The problem, as I see it, is that the characteristics (circles) on a Venn are selected by a person who may have biases. Their agenda then impacts everyone who sees the graph and assumes, like many people do when stuff looks scientific, that its conclusion is fact.

Perhaps our dialog and diagrams about alternative investments just need a little balance. Maybe we could even institute a five circle minimum. Regardless of the approach, it's clear we need a more (pun-intended) well-rounded approach.

A vector defining the edges, the first edge pointsfrom the first element to the second, the second edge from the thirdto the fourth, etc. For a numeric vector, these are interpretedas internal vertex ids. For character vectors, they are interpretedas vertex names.

For make_graph: extra arguments for the case when thegraph is given via a literal, see graph_from_literal.For directed_graph and undirected_graph:Passed to make_directed_graph or make_undirected_graph.

The number of vertices in the graph. This argument isignored (with a warning) if edges are symbolic vertex names. Itis also ignored if there is a bigger vertex id in edges. Thismeans that for this function it is safe to supply zero here if thevertex with the largest id is not an isolate.

make_graph can create some notable graphs. The name of thegraph (case insensitive), a character scalar must be supplied asthe edges argument, and other arguments are ignored. (A warningis given is they are specified.)

This is the smallest triangle-free graph that isboth 4-chromatic and 4-regular. According to the Grunbaum conjecture thereexists an m-regular, m-chromatic graph with n vertices for every m>1 andn>2. The Chvatal graph is an example for m=4 and n=12. It has 24 edges.

The Groetzschgraph is a triangle-free graph with 11 vertices, 20 edges, and chromaticnumber 4. It is named after German mathematician Herbert Groetzsch, and itsexistence demonstrates that the assumption of planarity is necessary inGroetzsch's theorem that every triangle-free planar graph is 3-colorable.

The Heawood graph is an undirected graph with 14 vertices and21 edges. The graph is cubic, and all cycles in the graph have six or moreedges. Every smaller cubic graph has shorter cycles, so this graph is the6-cage, the smallest cubic graph of girth 6.

A connected graph with 16 vertices and 27 edgescontaining no perfect matching. A matching in a graph is a set of pairwisenon-adjacent edges; that is, no two edges share a common vertex. A perfectmatching is a matching which covers all vertices of the graph.

The smallest hypotraceable graph, on 34 vertices and 52edges. A hypotraceable graph does not contain a Hamiltonian path but afterremoving any single vertex from it the remainder always contains aHamiltonian path. A graph containing a Hamiltonian path is called traceable.

Human vs. Lightning Caused Wildfires

Fire safety is an important idea to keep in mind both when visiting the park and at your home. The majority of wildfires are human caused. The graphs below were produced using statistics from the National Interagency Fire Center. The Southern Geographic Area is comprised of Texas, Oklahoma, Arkansas, Louisiana, Alabama, Mississippi, Florida, Georgia, Tennessee, North Carolina, South Carolina, Tennessee, Kentucky, Virginia and Puerto Rico. The first graph shows the number of wildfires started by lightning compared to the number of wildfires started by humans by year from 2001-2022 in the Southern Geographic Area. The second graph shows the number of wildfires started by lightning compared to the number of wildfires started by humans by year from 2001-2022 in the entire United States. Both graphs reveal there are significantly more wildfires started by humans than by lightning each year.

When camping in the park, especially in the remote campsites, make sure to park on pavement, or if pavement is not available, in short grass as this limits the risk that your vehicle may start a fire.

Burning of fuel with metal (ex: wooden pallets) is prohibited unless the metal parts (screws, nails, staples, etc.) are removed prior to burning. Burning of trash or other hazardous material is prohibited.

Keep an eye on the size of the fire, wind direction, and wind speed as these can increase the risk of a wildfire. Also keep in mind that low humidity, limited rainfall, and droughts are common in the region and greatly increase the wildfire risk.

Learn to be Firewise: Firewise is a way to prepare your house and your family for wildfires. Learn how to be Firewise. Being prepared for fire risks takes thought and preparation, but it can save your life, the lives of your loved ones, and your property and home. Keep these regulations and tips in mind to keep yourself safe.

Data Source for Graphs: National Interagency Fire Center. www.nifc.gov

In a recent post at the Energy Security and Freedom Substack, Thomas Shepstone asks a question dear to my heart: Why Are Electric Rates Higher Within Regional Transmission Organizations (RTO) Regions? Dysfunction Maybe?

The comments on his post are detailed and informative. Of course, I love reading books, and I even (sort of) enjoy writing them. Still, a book is not a conversation. A Substack post with a comment stream is a conversation!

Shepstone found the consumer prices for most of the states in the continental United States on the IEA site. He sorted the states into two groups (RTO and not-RTO) and then he made some tables of his results. He concluded that the RTO states have significantly higher consumer prices than the non-RTO states.

Not everyone who read the article agreed. One colleague emailed me to say that this sort of analysis needs some error bars. That colleague took the median (middle value) of each price set, and he also calculated the standard deviation for each price set. His chart appears below. The standard deviation bars overlap, which means there may not be a real difference between the two sets of data. (Due to job constraints, this colleague did not want his name used on this blog post.)

Steve Huntoon, a well-known energy lawyer who frequently writes for RTO Insider, wrote several comments on the Shepstone post. Huntoon suggested that I read a January 2023 blog post by James Bushnell of the Energy Institute of Haas:

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