Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

Godsil, Chris; Roberson, David E.; Rooney, Brendan; Šámal, Robert; Varvitsiotis, Antonios

doi:10.1007/s00454-017-9899-2

Cited by 4 publications

(19 citation statements)

References 23 publications

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“…The proof of Lemma 2.1 can be found in [5]. Next we show that max-rank optimal solutions for χ v (G) have the largest image among all optimal solutions.…”

Section: Characterizing Optimal Vector Coloringsmentioning

confidence: 80%

“…Of course, if one can show that no such matrix R exists, then this proves that the graph is uniquely vector colorable. This is the technique used to prove unique vector colorability in our previous works [5] and [6]. These works also present efficient algorithms for finding matrices R satisfying the conditions of the above theorem.…”

Section: The Gram Matrix Of An Optimal Vector Coloring Of G If and Onmentioning

confidence: 96%

“…It follows from results in [5] that any 1-walk-regular graph has a strictly complementary dual solution with strictly positive diagonal. As a consequence, if G is 1-walk-regular and H is connected with χ v (G) < χ v (H), then all of the optimal vector colorings of G × H are induced by G.…”

Section: Applicationsmentioning

confidence: 99%

“…However, we note that in the case of vector colorings the above theorem is actually stronger. This is because the theorem in [5] required the existence of a dual certificate (called a "spherical stress matrix"), and furthermore required that the max-rank vector coloring p be an optimal strict vector coloring as well. Since then we have discovered that these additional assumptions are superfluous and that the essential property of p is that it has maximum possible rank.…”

Section: The Gram Matrix Of An Optimal Vector Coloring Of G If and Onmentioning

confidence: 99%

“…Corollary 4.4 allows one to build many examples of uniquely vector colorable graphs. One could take G to be any of the Kneser or q-Kneser graphs, which were proven to be uniquely vector colorable in [5]. One could also let G be one of the Hamming graphs proven to be uniquely vector colorable in [6].…”

Section: Factors With Different Vector Chromatic Numbersmentioning

confidence: 99%

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Vector coloring the categorical product of graphs

et al. 2019

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A vector t-coloring of a graph is an assignment of real vectors p1, . . . , pn to its vertices such that p T i pi = t − 1 for all i = 1, . . . , n and p T i pj ≤ −1 whenever i and j are adjacent. The vector chromatic number of G is the smallest real number t ≥ 1 for which a vector t-coloring of G exists. For a graph H and a vector t-coloring p1, . . . , pn of a graph G, the assignment (i, ℓ) → pi is a vector t-coloring of the categorical product G × H. It follows that the vector chromatic number of G × H is at most the minimum of the vector chromatic numbers of the factors. We prove that equality always holds, constituting a vector coloring analog of the famous Hedetniemi Conjecture from graph coloring. Furthermore, we prove a necessary and sufficient condition for when all of the optimal vector colorings of the product can be expressed in terms of the optimal vector colorings of the factors. The vector chromatic number is closely related to the well-known Lovász theta function, and both of these parameters admit formulations as semidefinite programs. This connection to semidefinite programming is crucial to our work and the tools and techniques we develop could likely be of interest to others in this field.

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“…The proof of Lemma 2.1 can be found in [5]. Next we show that max-rank optimal solutions for χ v (G) have the largest image among all optimal solutions.…”

Section: Characterizing Optimal Vector Coloringsmentioning

confidence: 80%

Section: The Gram Matrix Of An Optimal Vector Coloring Of G If and Onmentioning

confidence: 96%

Section: Applicationsmentioning

confidence: 99%

Section: The Gram Matrix Of An Optimal Vector Coloring Of G If and Onmentioning

confidence: 99%

Section: Factors With Different Vector Chromatic Numbersmentioning

confidence: 99%

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Vector coloring the categorical product of graphs

et al. 2019

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An Annotated Glossary of Graph Theory Parameters, with Conjectures

et al. 2018

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Graph homomorphisms via vector colorings

Godsil

Roberson

Rooney

et al. 2019

European Journal of Combinatorics

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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.

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Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

Cited by 4 publications

References 23 publications

Vector coloring the categorical product of graphs

Vector coloring the categorical product of graphs

An Annotated Glossary of Graph Theory Parameters, with Conjectures

Graph homomorphisms via vector colorings

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