Lower bounds for the low hierarchy

Topological Lower Bounds for the Chromatic Number: A hierarchy

Topological lower bounds for the chromatic number: A hierarchy

Kneser’s conjecture, first proved by Lovász in 1978, states that the graph with all kelement subsets of {1, 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n−2k+2. Several other proofs have been published (by Bárány, Schrijver, Dol’nikov, Sarkaria, Kř́ıž, Greene, and others), all of them based on the Borsuk–Ulam theorem from algebraic topology, but otherwis...

Some lower bounds for the $L$-intersection number of graphs

‎For a set of non-negative integers~$L$‎, ‎the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$‎. ‎The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...

Some Lower Bounds for Estrada Index

Sum-of-Squares Hierarchy Lower Bounds for Symmetric Formulations

We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of “well-behaved” univariate polynomial inequalities. We illustrate the technique on two problems, one unconstrained and the other...