Is the Chromatic Number a Dirty Number?

Qi Hao spent half the day searching the library and finally found Brudno's paper.

The round trip between the school and the library took two hours, and getting lunch there was inconvenient. Qi Hao asked the librarian if he could take photos, and after receiving permission, he photographed the paper and brought it back to school.

Qi Hao seemed to be on a lucky streak. Over the next few days, he found two more papers.

Now only one reference remained missing. Since the other four all existed, Qi Hao believed the last one probably wasn't something the system had made up.

If he really couldn't find the last one, he would just have to let it go.

Only eighty days remained until the system task deadline. He had to hurry and digest Konrod's paper.

Qi Hao had no classes that afternoon, so he brought his secondhand laptop to the school library.

The used laptop wasn't much good for gaming, but it worked fine for reading papers, taking notes, and writing papers.

The library was packed today. Qi Hao figured it was because graduate entrance exam season was approaching.

He made two rounds before finding an empty seat.

"Sure enough, it's different now..."

Qi Hao opened the third section of Konrod's paper in the system. It dealt with partitioning the finite state space of game tree nodes.

But now, with Brudno's paper displayed on his laptop for reference, he finally understood the logic behind the state-space partitioning.

Konrod had used a clever equivalence-class construction here. He first classified the leaf nodes of the game tree by their terminal payoffs, then worked his way recursively up the levels of the tree, establishing local equivalence relations at each level.

In other words, he had broken a global problem of exponential complexity into local problems of polynomial complexity.

"So that's how it works."

Qi Hao wrote a note in the margin with his pencil, then turned to section five.

Then he got stuck.

At a crucial step in the proof, Konrod used the language of graph theory to describe the pruning process. He abstracted alpha-beta pruning into a reachability problem on a directed graph and introduced the concept of a dominating path.

Simply put, if a path from a root node to a leaf node was dominated at an intermediate node, then all subsequent nodes on that path could safely be pruned.

Qi Hao understood the idea, and it made sense logically.

But Konrod then used the chromatic number of a graph, the perfect graph theorem, and a lemma involving an upper bound on Ramsey numbers to prove the transitivity and completeness of the dominance relation. Qi Hao stared at the symbols, his mind blank.

What kind of number was a chromatic number?

He didn't even know the rigorous definition of the concept.

Surely numbers weren't called chromatic just because they were LSPs!

Qi Hao dismissed the absurd thought and looked through the references on his computer. One of them was a paper on fundamental theorems of perfect graphs and Ramsey theory. He tried to read it, but discovered that it in turn cited even more basic graph theory.

He was done for. At his normal learning pace, understanding all this would be more than a little difficult!

Subjects like graph theory and combinatorics weren't things a first-year student would get to study.

At that thought, Qi Hao began silently reciting, System, check my points.

[Current points: 150]

He tried typing "study graph theory" into the system search bar.

There were no results.

He searched for mathematical knowledge, but still got nothing.

Qi Hao was a little disappointed, though it was also what he had expected. This Soviet system never seemed to give him anything that would let him reach max level with one click. The papers he could obtain were already the most direct resources available.

He tried a different approach and browsed through the system store.

Function enhancements, auxiliary tools, information retrieval...

After flipping through a few pages, his gaze settled on an entry: [Deep Focus · Enhanced Learning State]

[Effect: Activates the brain's deep learning circuits, greatly increasing the efficiency of information absorption, comprehension, and internalization. Memory, logical reasoning, and pattern recognition are comprehensively enhanced for a limited time. Note: This ability does not directly provide any knowledge; it only enhances the learning process itself.

Cost: 50 points

Duration: 15 days

Cooldown: 30 days]

After reading the description, Qi Hao confirmed that this wouldn't download knowledge directly into his head. It would help him learn faster, remember better, and think more clearly.

It was like upgrading a computer's CPU.

He had 150 points. Spending 50 in exchange for fifteen days of accelerated learning...

Thinking of the papers he couldn't understand, Qi Hao hesitated no longer. "Redeem."

[Redemption successful! Deep Focus · Enhanced Learning State activated. Time remaining: 15 days, 00 hours, 00 minutes, 00 seconds]

[You may pause the timer at any time when you do not wish to use it.]

[Current points: 100]

In an instant, Qi Hao felt as if ice water had been poured over his brain. The sounds of pages turning and keyboards tapping in the library, and of someone making a phone call in the distance, were all shut out. His attention focused like a beam of light in the darkness.

Now he knew what he needed.

He got up and went upstairs to the mathematics collection on the fourth floor.

Qi Hao pulled three books from the shelves: Bondy and Murty's "Graph Theory with Applications," Liu Bailian's "Combinatorics," and Douglas West's "Introduction to Graph Theory."

Qi Hao stacked the three veritable bricks on the table.

For any undergraduate mathematics student, the nearly six-hundred-page classic textbook "Introduction to Graph Theory" alone was enough to torment them for a whole year.

The densely packed theorems and lemmas, along with graph-theory models that took immense mental effort to construct, usually required mountains of scratch paper and countless nights spent working until their eyes turned red, just to barely get through them.

As for "Combinatorics," that was the hairline killer that struck fear into the hearts of countless science students.

An ordinary person who could thoroughly master seven or eight pages of a book like that in a day and work through the exercises afterward would already be considered smart.

But what Qi Hao felt at that moment was unlike anything he had ever experienced.

He opened a book, his eyes moving steadily across the pages. Isomorphic graphs, adjacency matrices, spanning trees, and complex generating-function expansions, which normally required repeated work with pen and paper, now materialized on their own in his mind.

The complex network nodes and edges, originally static, projected themselves in his mind and then broke apart and recombined. Complex algebraic calculations began automatically in his head, with no need for external tools to record intermediate variables!

And now, at last, he knew that a chromatic number was the minimum number of colors needed to classify a group of things and distinguish them by color, provided that related, adjacent things could never be placed in the same category.

A simple example: any two provinces right next to each other on a map absolutely couldn't be colored the same color, or you wouldn't be able to tell where the boundary between them was. If you wanted to save money by buying the fewest watercolor markers needed to color the entire map, what was the minimum number of colors you would have to buy?

That "minimum number of colored markers you needed to buy" was what mathematics called the chromatic number of that map.

More complexly, chromatic numbers could be applied in the abstract to conflict-free scheduling in countless real-world systems, such as radio frequency allocation.

In a city dense with communications base stations, neighboring stations would interfere with each other if they used the same frequency. To save costs, operators would naturally want to cover the whole city with the fewest frequency resources without causing interference. The answer sought here was still a chromatic number.

So chromatic numbers were perfectly respectable and had nothing to do with LSPs.

Sitting across from Qi Hao was a graduate student losing his hair over the proposal for his master's thesis. At first, he hadn't paid attention to the person who had casually grabbed three god-tier books, until he heard the regular sound of pages turning across from him.

It was easy to get distracted when writing a thesis, especially for someone like him who didn't know what to write.

He noticed that Qi Hao turned a page roughly every minute or so.

A page a minute while reading "Introduction to Graph Theory"?

At that speed, he wouldn't even have time to read all the formulas aloud, would he? He didn't even take out a sheet of scratch paper, just sat there staring at a math book?

Was this guy planning to take a photo in the library and post it to his Moments, or was he pretending to be a top student to attract some junior girl's attention?

Hmph. There's actually someone who puts on an even bigger show than I do.

What's the point of showing off like that, other than getting a junior girl's admiration?

None of the knowledge will make it into his head!

He shook his head, thinking that when the guy had to write a thesis, he would know what suffering was!