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Any two reduced expressions for the same Coxeter group element are related by a sequence of commutation and braid moves. We say that two reduced expressions are braid equivalent if they are related via a sequence of braid moves, and the corresponding equivalence classes are called braid classes. Each braid class can be encoded in terms of a braid graph in a natural way. In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are cubical by constructing an embedding of the braid graph into a hypercube graph whose image is an induced subgraph of the hypercube. For a special class of links, called Fibonacci links, we prove that this embedding is an isometry from the corresponding braid graph to a Fibonacci cube graph.
For all $nge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by ErdH{o}s, and extends results by Grzesik and Hatami, Hladky, Kr{
Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is colourful if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph con
An orientation of a graph is semi-transitive if it is acyclic, and for any directed path $v_0rightarrow v_1rightarrow cdotsrightarrow v_k$ either there is no arc between $v_0$ and $v_k$, or $v_irightarrow v_j$ is an arc for all $0leq i<jleq k$. An un
In 1967, ErdH{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of ErdH{o}s and Hajnal together with Shearers classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows tha
It is known that the recently discovered representations of the Artin groups of type A_n, the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D_n and E_n which also lead to the newly found faithful representat