The Gram dimension \rm gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in ℝk, having the same inner products on the edges of the graph. The class of graphs satisfying \rm gd(G) ≤ k is minor closed for fixed k, so it can characterized by a finite list of forbidden minors. For k ≤ 3, the only forbidden minor is Kk + 1. We show that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly [5,6].
|Title of host publication||Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012)|
|Editors||A.R. Mahjoub, V. Markakis, I. Milis, V. Paschos|
|Place of Publication||Berlin Heidelberg|
|Publication status||Published - 2012|
|Name||Lecture Notes in Computer Science|
Laurent, M., & Varvitsiotis, A. (2012). The gram dimension of a graph. In A. R. Mahjoub, V. Markakis, I. Milis, & V. Paschos (Eds.), Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012) (Vol. 7422, pp. 356-367). (Lecture Notes in Computer Science; Vol. 7422). Springer Verlag. http://link.springer.com/chapter/10.1007/978-3-642-32147-4_32