Vector coloring the categorical product of graphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10453969" target="_blank" >RIV/00216208:11320/20:10453969 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=QiujmBioxl" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=QiujmBioxl</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s10107-019-01393-0" target="_blank" >10.1007/s10107-019-01393-0</a>

Result language
angličtina
Original language name
Vector coloring the categorical product of graphs
Original language description
A vector t-coloring of a graph is an assignment of real vectors p1, ... , pn 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 p1, ... , pn of G, the map taking (i, ℓ) ELEMENT OF V(G) x V(H) to pi is a vector t-coloring of the categorical product Gx H. It follows that the vector chromatic number of Gx 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 Gx 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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project
<a href="/en/project/GA16-19910S" target="_blank" >GA16-19910S: Graphs and mappings -- Algebraic properties of graphs</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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