On the product dimension of clique factors

Noga Alon, Ryan Alweiss

Research output: Contribution to journalArticlepeer-review


The product dimension of a graph G is the minimum possible number of proper vertex colorings of G so that for every pair u,v of non-adjacent vertices there is at least one coloring in which u and v have the same color. What is the product dimension Q(s,r) of the vertex disjoint union of r cliques, each of size s? Lovász, Nešetřil and Pultr proved in 1980 that for s=2 it is (1+o(1))log2r and raised the problem of estimating this function for larger values of s. We show that for every fixed s, the answer is still (1+o(1))log2r where the o(1) term tends to 0 as r tends to infinity, but the problem of determining the asymptotic behavior of Q(s,r) when s and r grow together remains open. The proof combines linear algebraic tools with the method of Gargano, Körner, and Vaccaro on Sperner capacities of directed graphs.

Original languageEnglish
Article number103097
JournalEuropean Journal of Combinatorics
StatePublished - May 2020


FundersFunder number
National Science FoundationDMS-1855464, 1855464
German-Israeli Foundation for Scientific Research and DevelopmentG-1347-304.6/2016
Israel Science Foundation281/17


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