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Conic formulations of graph homomorphisms

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Abstract

Given graphs X and Y, we define two conic feasibility programs which we show have a solution over the completely positive cone if and only if there exists a homomorphism from X to Y. By varying the cone, we obtain similar characterizations of quantum/entanglement-assisted homomorphisms and three previously studied relaxations of these relations. Motivated by this, we investigate the properties of these “conic homomorphisms” for general (suitable) cones. We also consider two generalized versions of the Lovász theta function, and how they interact with these conic homomorphisms. We prove analogs of several results on classical graph homomorphisms as well as some monotonicity theorems. We also show that one of the generalized theta functions is multiplicative on lexicographic and disjunctive graph products.

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Notes

  1. See Sect. 10.2 for a brief description of quantum strategies, or see Sects. 5 and 6 of [8] for a more detailed analysis.

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Acknowledgments

D. E. Roberson is supported in part by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13.

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  1. Nanyang Technological University and Centre for Quantum Technologies, Singapore, Singapore

    David E. Roberson

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  1. David E. Roberson
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Roberson, D.E. Conic formulations of graph homomorphisms. J Algebr Comb 43, 877–913 (2016). https://doi.org/10.1007/s10801-016-0665-y

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  • DOI: https://doi.org/10.1007/s10801-016-0665-y

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