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Taken from: [1, Exercise 12.2]. Exercise 12.2 Prove the assertion of the Note following Definition 12.10. That is, let $\Delta$ be a pure $(d-1)$-dimensional simplicial complex. Then a facet ordering $F_1, \ldots, F_t$ is a shelling if and only if for all $2 \leq i \leq t$, the subcomplex $\left\langle F_1, \ldots, F_{i-1}\right\rangle \cap\le...

Taken from: [1, Exercise 9.7]. Exercise 9.7 (a) Let $ G $ be a bipartite graph with vertex bipartition $ (A, B) $. Suppose that $ \deg v = a $ for all $ v \in A $, and $ \deg v = b $ for all $ v \in B $. Let $ \boldsymbol{A} $ and $ \boldsymbol{L} $ denote the adjacency matrix and Laplacian matrix of $ G $, respectively. Show that if the eigenv...

Spanish version Hola a todxs, desde cualquier parte del mundo. He calculado que el número de formas de colorear el mapa de Colombia sin que dos regiones adyacentes compartan color, usando solamente 4 colores, es: \[\Large{283\ 115\ 520}\] Cabe anotar que he considerado a Bogotá como un departamento aislado, y el archipiélago de San Andrés y P...