science

Quartic graph

Quartic graph is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Quartic graph rather than just read about it. In short: In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph.

Key takeaways

Quartic graph belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quartic graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quartic graph from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph.

Examples

Several well-known graphs are quartic. They include:

The complete graph K 5 {\displaystyle K_{5}} , a quartic graph with 5 vertices, is the smallest possible quartic graph. The octahedral graph K 2 , 2 , 2 {\displaystyle K_{2,2,2}} , the graph of the regular octahedron, has 6 vertices. The Chvátal graph, another quartic graph with 12 vertices, is the smallest quartic graph that both has no triangles and cannot be colored with three colors. The Folkman graph, a quartic graph with 20 vertices, is the smallest semi-symmetric graph. This means that it has symmetries taking every two edges to each other, but does not have symmetries taking every two vertices to each other. The Meredith graph, a quartic graph with 70 vertices, is 4-connected but has no Hamiltonian cycle, disproving a conjecture of Crispin Nash-Williams. Every medial graph is a quartic plane graph, and every quartic plane graph is the medial graph of a pair of dual plane graphs or multigraphs. Knot diagrams and link diagrams are also quartic plane multigraphs, in which the vertices represent the crossings of the diagram and are marked with additional information concerning which of the two branches of the knot crosses the other branch at that point. The line graph of any cubic graph is quartic; the cuboctahedral graph is an example, as the line graph of a cube graph. The subdivided double of a quartic graph is another, larger, quartic graph. It is obtained by subdividing each edge of the given graph into a two-edge path, and then replacing each vertex that came from the given graph by a pair of vertices, both adjacent to the same neighborhood of subdivision vertices. For instance, the Folkman graph is the subdivided double of the complete graph K 5 {\displaystyle K_{5}} .

Properties Because the degree of every vertex in a quartic graph is even, every connected quartic graph has an Euler tour. And as with regular bipartite graphs more generally, every bipartite quartic graph has a perfect matching. In this case, a much simpler and faster algorithm for finding such a matching is possible than for irregular graphs: by selecting every other edge of an Euler tour, one may find a 2-factor, which in this case must be a collection of cycles, each of even length, with each vertex of the graph appearing in exactly one cycle. By selecting every other edge again in these cycles, one obtains a perfect matching in linear time. The same method can also be used to color the edges of the graph with four colors in linear time. Quartic graphs have an even number of Hamiltonian decompositions.

Open problems It is an open conjecture whether all quartic Hamiltonian graphs have an even number of Hamiltonian circuits, or have more than one Hamiltonian circuit. The answer is known to be false for quartic multigraphs.

Worked examples

Example 1 — a first encounter with Quartic graph

Start with the simplest possible case. Write down what Quartic graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quartic graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quartic graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quartic graph

In research: Quartic graph appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quartic graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quartic graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph families, Regular graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Quartic graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Quartic graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quartic graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quartic graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quartic graph in simple terms?

In the mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph.

Why does Quartic graph matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quartic graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quartic graph.