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Displaying similar documents to “Fixed simplex property for retractable complexes.”

On 3-simplicial vertices in planar graphs

Endre Boros, Robert E. Jamison, Renu Laskar, Henry Martyn Mulder (2004)

Discussiones Mathematicae Graph Theory

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A vertex v in a graph G = (V,E) is k-simplicial if the neighborhood N(v) of v can be vertex-covered by k or fewer complete graphs. The main result of the paper states that a planar graph of order at least four has at least four 3-simplicial vertices of degree at most five. This result is a strengthening of the classical corollary of Euler's Formula that a planar graph of order at least four contains at least four vertices of degree at most five.

An equivalence criterion for 3-manifolds.

M. R. Casali (1997)

Revista Matemática de la Universidad Complutense de Madrid

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Within geometric topology of 3-manifolds (with or without boundary), a representation theory exists, which makes use of 4-coloured graphs. Aim of this paper is to translate the homeomorphism problem for the represented manifolds into an equivalence problem for 4-coloured graphs, by means of a finite number of graph-moves, called dipole moves. Moreover, interesting consequences are obtained, which are related with the same problem in the n-dimensional setting.

On the simplex graph operator

Bohdan Zelinka (1998)

Discussiones Mathematicae Graph Theory

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A simplex of a graph G is a subgraph of G which is a complete graph. The simplex graph Simp(G) of G is the graph whose vertex set is the set of all simplices of G and in which two vertices are adjacent if and only if they have a non-empty intersection. The simplex graph operator is the operator which to every graph G assigns its simplex graph Simp(G). The paper studies graphs which are fixed in this operator and gives a partial answer to a problem suggested by E. Prisner.