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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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      67
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      6answers
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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 ...
      63
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      4answers
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      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,...
      62
      votes
      9answers
      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 ...
      56
      votes
      4answers
      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 ...
      54
      votes
      17answers
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      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 ...
      53
      votes
      4answers
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      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 ...
      49
      votes
      1answer
      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!] ...
      48
      votes
      4answers
      9k views

      Do there exist chess positions that require exponentially many moves to reach?

      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\...
      46
      votes
      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 ...
      44
      votes
      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 ...
      43
      votes
      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, ...
      43
      votes
      3answers
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      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 ...
      42
      votes
      6answers
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      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 ...
      41
      votes
      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 ...
      40
      votes
      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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