science

Goldberg–Seymour conjecture

Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture rather than just read about it. In short: In graph theory, the Goldberg–Seymour conjecture states that, for a multigraph G {\displaystyle G} χ ′ ⁡ ( G ) ≤ max ( 1 + Δ ⁡ ( G ) , Γ ⁡ ( G ) ) {\displaystyle \operatorname {\chi '} (G)\leq \max(1+\operatorname {\Delta } (G),\,\operatorname {\Gamma } (G))} where χ ′ ⁡ ( G ) {\displaystyle \operatorname {\chi '} (G)} is the edge chromatic number of G, Δ ⁡ ( G ) {\displaystyle \operatorname {\Delta } (G)} is its ma…

Key takeaways

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

Reference excerpt

In graph theory, the Goldberg–Seymour conjecture states that, for a multigraph G {\displaystyle G}

χ ′ ⁡ ( G ) ≤ max ( 1 + Δ ⁡ ( G ) , Γ ⁡ ( G ) ) {\displaystyle \operatorname {\chi '} (G)\leq \max(1+\operatorname {\Delta } (G),\,\operatorname {\Gamma } (G))}

where χ ′ ⁡ ( G ) {\displaystyle \operatorname {\chi '} (G)} is the edge chromatic number of G, Δ ⁡ ( G ) {\displaystyle \operatorname {\Delta } (G)} is its maximum degree, and

Γ ⁡ G = max H ⊂ G | E ( H ) | ⌊ 1 2 | V ( H ) | ⌋ . {\displaystyle \operatorname {\Gamma } G=\max _{H\subset G}{\frac {|E(H)|}{\lfloor {\frac {1}{2}}|V(H)|\rfloor }}.}

This above quantity is twice the arboricity of G. It is sometimes called the density of G. Here, G can be a multigraph and can have loops. For simple graphs, this result follows from Vizing's theorem.

Background It is already known that for loopless G (but can have parallel edges):

χ ′ ⁡ G ≥ max ( Δ ⁡ G , ⌈ Γ ⁡ G ⌉ ) . {\displaystyle \operatorname {\chi '} G\geq \max(\operatorname {\Delta } G,\lceil \operatorname {\Gamma } G\rceil ).}

When does equality not hold? It does not hold for the Petersen graph. It is hard to find other examples. It is currently unknown whether there are any planar graphs for which equality does not hold. This conjecture is named after Mark K. Goldberg of Rensselaer Polytechnic Institute and Paul Seymour of Princeton University, who arrived to it independently of Goldberg.

Announced proof In 2019, an alleged proof was announced by Chen, Jing, and Zang in the paper. Part of their proof was to find a suitable generalization of Vizing's theorem (which says that for simple graphs χ ′ ⁡ G ≤ 1 + Δ ⁡ G {\displaystyle \operatorname {\chi '} G\leq 1+\operatorname {\Delta } G} ) to multigraphs. In 2023, Jing announced a new proof with a polynomial-time edge coloring algorithm achieving the conjectured bound.

See also Petersen graph#Coloring Fractional coloring Graph coloring

References

Worked examples

Example 1 — a first encounter with Goldberg–Seymour conjecture

Start with the simplest possible case. Write down what Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture

In research: Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture 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: Goldberg–Seymour conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Conjectures, Graph coloring, so understanding it makes those chapters shorter.
In everyday life: Look for Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Goldberg–Seymour conjecture in simple terms?

In graph theory, the Goldberg–Seymour conjecture states that, for a multigraph G {\displaystyle G} χ ′ ⁡ ( G ) ≤ max ( 1 + Δ ⁡ ( G ) , Γ ⁡ ( G ) ) {\displaystyle \operatorname {\chi '} (G)\leq \max(1+\operatorname {\Delta } (G),\,\operatorname {\Gamma } (G))} where χ ′ ⁡ ( G ) {\displaystyle \opera…

Why does Goldberg–Seymour conjecture 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 Goldberg–Seymour conjecture?

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 Goldberg–Seymour conjecture.