Vector coloring the categorical product of graphs

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Vector coloring the categorical product of graphs

Popis výsledku v původním jazyce

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&lt;=-1, whenever i and j are adjacent. The vector chromatic number of G is the smallest number t&gt;= 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.

Název v anglickém jazyce

Vector coloring the categorical product of graphs

Popis výsledku anglicky

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&lt;=-1, whenever i and j are adjacent. The vector chromatic number of G is the smallest number t&gt;= 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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GA16-19910S" target="_blank" >GA16-19910S: Grafy a zobrazení -- Algebraické vlastnosti grafů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Mathematical Programming Computation

ISSN

1867-2949

e-ISSN

1867-2957

Svazek periodika

182

Číslo periodika v rámci svazku

1-2

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

40

Strana od-do

275-314

Kód UT WoS článku

000542402700009

EID výsledku v databázi Scopus

2-s2.0-85064644851