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Here's a proof I concocted, utilising our favourite graph: Consider the Petersen graph $P$, obtained from the dodecahedral graph $D$ by quotienting out by the antipodal map $\theta : D \rightarrow P$ …

Let $\mathcal{G}$ be the set of isomorphism classes of finite groups. There is an operation $\mathrm{Aut} : \mathcal{G} \rightarrow \mathcal{G}$ which gives the automorphism group of a given group, u …

The Leech lattice $\Lambda_{24}$ answers your question with quite a large multiplicity. The unique $24$-dimensional laminated lattice, defined by $\Lambda_0$ being the one-point lattice and $\Lambda …

I get a total of $62$ by appealling to Burnside's lemma. In particular, note that the objects we are counting are 2-colourings of the vertices of a decagon up to: Rotations; Reflections composed wit …

I can, at least, answer your question 'Is this in fact the correct answer?' with an affirmative 'no'. Specifically, we can replace the upper bound $2^{n/3} \approxeq 1.26^n$ with the slightly better …

In Andreas Blass's famous paper `Seven Trees in One', the existence of a natural bijection between binary trees and 7-tuples of binary trees is related to the equation $T^7 = T$ being satisfied by a c …

As the graph is self-complementary, by definition there exists some isomorphism $f$ from the graph to its complement. Consider $f$ as a bijection from the vertex-set of the graph to itself (i.e. a per …

EDIT: This answer pertains to a misinterpretation of the question before it was later clarified, so does not answer the question as currently stated: Two distinct 4-cycles share a 3-vertex path if …

I've been pondering the following generalisation of a famous problem (the special case where $T = \mathbb{N})$: Question: We have some totally-ordered set $T$ of mathematicians, each wearing a hat wh …

Note that if a graph is vertex-transitive, then it is `symmetric' by symmetry and we can wlog assume $i = 1$. The skeleton of the truncated icosidodecahedron is a counter-example to your conjecture, s …

Let $\mathbb{F}_q$ be a finite field and $k$ be a positive integer. If we colour each point of the infinite-dimensional projective space $\mathbb{F}_q \mathbb{P}^{\infty}$ with one of $k$ colours, can …

After further investigation, it appears that disappointingly the answer is 'no', and a proof appears in Lemma 2.4 of Partition Theorems for Subspaces of Vector Spaces (Cates and Hindman, 1975).

I can prove that all sufficiently large integers are representable. Firstly, observe that there is a unique way, up to isomorphism, to choose three edges $p, q, r$ of the Petersen graph such that the …

Fedja has already answered one possible interpretation of your question in a comment, where the common ratio is required to be an integer. Here's further explanation, together with a multiplicative an …

The Held-Karp algorithm for TSP can be modified to count Hamiltonian cycles. The idea is to fix a starting vertex $v \in V$ and inductively count, for every vertex $w \neq v$ and subset $S \subseteq V …