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Questions tagged [graph-theory]

Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.

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Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another. My main ...
9k views

What are the implications of the new quasi-polynomial time solution for the Graph Isomorphism problem?

This week, news came out that Laszlo Babai has found a quasi-polynomial time algorithm to solve the Graph Isomorphism problem (that is: $O(\exp(polylog(n)))$). He is giving a series of talks this week,...
6k views

What is a continuous path?

I would like some help, because I am getting mad trying to answer the following Question: Let $X$ be a topological space, what is a continuous path in $X$? Well, maybe you're already getting ...
13k views

What is a chess piece mathematically?

Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained open for more ...
8k views

Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors. The purpose of this question is to collect generalizations, variations, and strengthenings of the four ...
8k views

Connectivity of the Erd?s–Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erd?s–Rényi random graph $G(n,p)$ is ...
4k views

Why are there 1024 Hamiltonian cycles on an icosahedron?

Fix one edge $e$ of the graph (1-skeleton) of an icosahedron. By a computer search, I found that there are 1024 Hamiltonian cycles that include $e$. [But see edit below re directed vs. undirected!] ...
9k views

By "chess" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an $n\... 3answers 4k views Does knight behave like a king in his infinite odyssey? The Knight's Tour is a well-known mathematical chess problem. There is an extensive amount of research concerning this question in two/higher dimensional finite boards. Here, I would like to tackle ... 0answers 1k views Does every triangle-free graph with maximum degree at most 6 have a 5-colouring? A very specific case of Reed's Conjecture Reed's$\omega$,$\Delta$,$\chi$conjecture proposes that every graph has$\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here$\chi$is the chromatic ... 4answers 7k views Why are planar graphs so exceptional? As compared to classes of graphs embeddable in other surfaces. Some ways in which they're exceptional: Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, ... 3answers 2k views History of the four-colour problem It is stated in many places that the first published reference to the four-colour problem (aka the four-color problem) was an anonymous article in The Athen?um of April 14, 1860, attributed to de ... 6answers 4k views Is it easy to produce hard-to-color graphs? This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ... 1answer 1k views Can't one walk to infinity on the prime numbers with finitely many distinct affine steps? Let$(a_1,b_1), \dots, (a_k,b_k)$be finitely many pairs of positive integers, and let$\Gamma$be the graph whose vertices are the prime numbers and in which two vertices$p$and$q$are connected by ... 15answers 7k views Strengthening the Induction Hypothesis Suppose you are trying to prove result$X$by induction and are getting nowhere fast. One nice trick is to try to prove a stronger result$X'\$ (that you don't really care about) by induction. This ...

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