## Abstract

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↦p _{i} to its vertices such that 〈p _{i} ,p _{j} 〉≤−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 K _{n:r} and the q-Kneser graph qK _{n:r} are cores, and furthermore, that for n∕r=n ^{′} ∕r ^{′} there exists a homomorphism K _{n: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 (http://www.maths.gla.ac.uk/ es/srgraphs.php)and found that at least 84% are cores.

Original language | English |
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Journal | European Journal of Combinatorics |

Volume | 79 |

Pages (from-to) | 244-261 |

ISSN | 0195-6698 |

DOIs | |

Publication status | Published - 1 Jun 2019 |