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visits | member for | 3 years |
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stats | profile views | 262 |
Mar 25 |
comment |
Isometries of some simple Cayley graphs
Sorry, I missed the "finite". Please disregard my comment. |
Mar 24 |
awarded | Yearling |
Mar 23 |
comment |
Isometries of some simple Cayley graphs
Suppose that we take as connecting set the whole group except for the identity. Then the graph is complete and its automorphism group is the full symmetric group. This is always bigger than the group generated by translations and group automorphisms once |G|>4. In other words, except for a few small exceptions, the answer cannot be yes for all generating sets of a given group. |
Mar 11 |
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Generalization of Hamiltonian cycle
Sure, the equivalences don't necessarily hold for infinite graphs, I was simply answering the remark in the OP that there isn't an obvious way to generalise HC. |
Mar 11 |
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Generalization of Hamiltonian cycle
Moreover, I think Hamiltonian cycles generalise in a straightforward way to infinite graphs. After all, a HC is simply a 2-valent spanning subgraph. This works in the infinite case as well and is well-accepted I think. |
Mar 11 |
answered | Generalization of Hamiltonian cycle |
Mar 11 |
comment |
Section of Cayley graphs
Another way to say what Dave said is that, since $S$ is the full preimage of $S_1$, as a graph, $\mathrm{Cay}(G,S)$ is the lexicographic product of $\mathrm{Cay}(G_1,S_1)$ with the edgeless graph on $G/G_1$ (and thus any choice of one vertex per preimage will induce an embedding). |
Feb 21 |
awarded | Revival |
Feb 6 |
answered | Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$ |
Jan 30 |
comment |
Automorphism group of regular graph
@Bredan, note that here $n$ is the order of the graph. |
Jan 23 |
comment |
Is it possible to create an infinite sequence in which no subsequence is repeated 3 times in a row?
Furthermore, to be a bit pedantic, there is nothing "illegal" about a triple repetition in chess, it's simply that any of the player can claim a draw, but they can also both decide to play on. |
Jan 23 |
comment |
Is it possible to create an infinite sequence in which no subsequence is repeated 3 times in a row?
This is not a mathematical comment but a chess one (and Steven Landsburg has already hinted at this): the opening post misunderstands and misquotes the threefold repetition rule in chess. A player can claim a draw if the same POSITION occurs three times, the sequence of moves if irrelevant (it says just as much on the linked page). As there are only finitely many possible positions, there are only finitely many games that avoid a triple repetition. The mathematical question asked thus has nothing to do with the chess one. |
Jan 22 |
answered | Permutation Group Question |
Jan 22 |
awarded | Scholar |
Jan 22 |
revised |
Vertex-primitive graphs with two vertices having almost the same neighbourhood
deleted 3846 characters in body |
Jan 22 |
accepted | Vertex-primitive graphs with two vertices having almost the same neighbourhood |
Jan 22 |
answered | Vertex-primitive graphs with two vertices having almost the same neighbourhood |
Jan 15 |
comment |
What is the smallest 4-chromatic graph of girth 5?
So 21 is the answer then? Is the Brinkman graph the unique 4-chromatic graph of girth 5 and order 21? |
Jan 6 |
comment |
Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph
Thanks Chris. By the way, Ashwin, since you asked for it, here is a specific example with L_v of order 3. It is a 4-valent 2-arc-transitive graph of order 32. The vertex-stabiliser has order 72: jan.ucc.nau.edu/swilson/C4Site/N032/N032i005/Forms.html |
Jan 6 |
comment |
Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph
@Ashwin, You are correct that Chris was referring to "normal" Cayley graphs with the comment at the end of his answer. While it is likely that, in some sense, "most" Cayley graphs are normal (which is why he said "tends to"), there are still plenty of Cayley examples to your question. In fact, if I remember correctly, for any starting vertex-transitive graph $\Gamma$, there are infinitely many values of $k$ for which the construction in my previous comment will yield a Cayely graph. |