Vector coloring the categorical product of graphs

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

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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A vector t-coloring of a graph is an assignment of real vectors p 1 , … , p n to its vertices such that piTpi=t-1, for all i= 1 , … , n and piTpj≤-1, whenever i and j are adjacent. The vector chromatic number of G is the smallest number t≥ 1 for which a vector t-coloring of G exists. For a graph H and a vector t-coloring p 1 , … , p n of G, the map taking (i, ℓ) ∈ V(G) × V(H) to p i 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 necessary and sufficient conditions under which all optimal vector colorings of G× H are induced by optimal vector colorings of the factors. Our proofs rely on various semidefinite programming formulations of the vector chromatic number and a theory of optimal vector colorings we develop along the way, which is of independent interest.

Original languageEnglish
JournalMathematical Programming
Number of pages40
Publication statusPublished - 1 Jan 2019


  • Categorical graph product
  • Hedetniemi’s conjecture
  • Lovász ϑ number
  • Semidefinite programming
  • Vector coloring


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