论文信息 - Geometrical embeddings of graphs

Geometrical embeddings of graphs

The scalar product dimension d ( G ) of a graph G is defined to be the minimum number m such that the vertices x of G can be represented by vectors x¯∈ R M with the property that xy is an edge of G iffx¯y¯⩾ t for some real threshold t . Graphs G with d(G) = 1 are characterized and d ( G ) is determined for a variety of graphs. The dimension is compared with the (related and well-known) threshold dimension. Two other variants (spherical and distance dimensions) are considered. Upper bounds for graphs with small maximum degree (or with small maximum degree of the complement) are established.

[1] David G. Larman. A Triangle Free Graph Which Cannot be square root of 3 Imbedded in any Euclidean Unit Sphere , 1978, J. Comb. Theory, Ser. A.

[2] A. Hajnal,et al. On decomposition of graphs , 1967 .

[3] Brian Alspach,et al. On embedding triangle-free graphs in unit spheres , 1977, Discret. Math..

[4] M. Simonovits,et al. ON THE CHROMATIC NUMBER OF GEOMETRIC GRAPHS , 1980 .

[5] M. Yannakakis. The Complexity of the Partial Order Dimension Problem , 1982 .

[6] Janos Simon,et al. Probabilistic Communication Complexity , 1986, J. Comput. Syst. Sci..

[7] Vojtech Rödl. On combinatorial properties of spheres in euclidean spaces , 1984, Comb..

[8] P. Hammer,et al. Aggregation of inequalities in integer programming. , 1975 .

[9] Peter Frankl,et al. Embedding the n-cube in Lower Dimensions , 1986, Eur. J. Comb..

[10] Hiroshi Maehara,et al. Space graphs and sphericity , 1984, Discret. Appl. Math..

[11] László Lovász,et al. On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[12] Moshe Rosenfeld. Triangle free graphs that are not 3-embeddable in Sd , 1982 .

[13] M. Golumbic. Algorithmic graph theory and perfect graphs , 1980 .

[14] W. T. Tutte,et al. ON THE DIMENSION OF A GRAPH , 1965 .

[15] Vojtech Rödl,et al. Geometrical realization of set systems and probabilistic communication complexity , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).