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123266 Discrete Comput Geom (2017) 58:265-292 In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjacency matrix of G, which we call least eigenvalue frameworks. We identify two necessary and sufficient conditions for such frameworks to be universally completable. Our conditions also allow us to give algorithms for determining whether a least eigenvalue framework is universally completable. Furthermore, our computations for Cayley graphs on Z n 2 (n ≤ 5) show that almost all of these graphs have universally completable least eigenvalue frameworks. In the second part of this work we study uniquely vector colorable (UVC) graphs, i.e., graphs for which the semidefinite program corresponding to the Lovász theta number (of the complementary graph) admits a unique optimal solution. We identify a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework. This allows us to prove that Kneser and q-Kneser graphs are UVC. Lastly, we show that least eigenvalue frameworks of 1-walk-regular graphs always provide optimal vector colorings and furthermore, we are able to characterize all optimal vector colorings of such graphs. In particular, we give a necessary and sufficient condition for a 1-walk-regular graph to be uniquely vector colorable.

The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane's Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney's Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).

In this paper we study the existence of homomorphisms G → H using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number t ≥ 2 for which there exists an assignment of unit vectors i → pi to its vertices such that pi, pj ≤ −1/(t − 1), when i ∼ j. Our approach allows to reprove, without using the Erdős-Ko-Rado Theorem, that for n > 2r the Kneser graph Kn:r and the q-Kneser graph qKn:r are cores, and furthermore, that for n/r = n ′ /r ′ there exists a homomorphism Kn:r → K n ′ :r ′ if and only if n divides n ′ . In terms of new applications, we show that the even-weight component of the distance k-graph of the n-cube H n,k is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms H n,k → H n ′ ,k ′ when n/k = n ′ /k ′ . Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence's list of strongly regular graphs [25] and found that at least 84% are cores.