arXiv subjects
Publications and source records attributed to Hamed Hatami.
For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l^1(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We show that there is no commutative semilattice with amenability constant between 5 and 9.
Circular chromatic number, $χ_c$ is a natural generalization of chromatic number. It is known that it is \NP-hard to determine whether or not an arbitrary graph $G$ satisfies $χ(G) = χ_c(G)$. In this paper we prove that this problem is \NP-hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers $k \ge 2$ and $n \ge 3$, for a given graph $G$ with $χ(G)=n$, it is \NP-complete to verify if $χ_c(G) \le n- \frac{1}{k}$.
Suppose we have a family ${\cal F}$ of sets. For every $S \in {\cal F}$, a set $D \subseteq S$ is a {\sf defining set} for $({\cal F},S)$ if $S$ is the only element of $\cal{F}$ that contains $D$ as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares. In this paper, first, we propose the concept of a defining set of a logical formula, and we prove that the computational complexity of such a problem is $Σ_2$-complete. We also show that the computational complexity of the following problem about the defining set of vertex colorings of graphs is $Σ_2$-complete: {\sc Instance:} A graph $G$ with a vertex coloring $c$ and an integer $k$. {\sc Question:} If ${\cal C}(G)$ be the set of all $χ(G)$-colorings of $G$, then does $({\cal C}(G),c)$ have a defining set of size at most $k$? Moreover, we study the computational complexity of some other variants of this problem.
An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree $Δ$ and with no isolated edges has an \avd-coloring with at most $Δ+300$ colors, provided that $Δ>10^{20}$.
We prove that a random cubic graph almost surely is not homomorphic to a cycle of size 7. This implies that there exist cubic graphs of arbitrarily high girth with no homomorphisms to the cycle of size 7.
A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. The cardinality of the largest critical set in any Latin square of order $n$ is denoted by $\lcs{n}$. We give a lower bound for $\lcs{n}$ by showing that $\lcs{n} \geq n^2(1-\frac{2 + \ln 2}{\ln n})+n(1+\frac {\ln (8 π)} {\ln n})-\frac{\ln 2}{\ln n}.$
A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. For a Latin square $L$, $\scs{L}$ denotes the size of the smallest critical set of $L$, and $\scs{n}$ is the minimum of $\scs{L}$ over all Latin squares $L$ of order $n$. We find an upper bound for the number of partial Latin squares of size $k$ and prove that $$n^2-(e+o(1))n^{10/6} \le \max \scs{L} \le n^2-\frac{\sqrtπ}{2}n^{9/6}.$$ % This improves a result of N. Cavenagh (Ph.D. thesis, The University of Queensland, 2003) and disproves one of his conjectures. Also it improves the previously known lower bound for the size of the largest critical set of any Latin square of order $n$.
This paper proves that if $G$ is a graph (parallel edges allowed) of maximum degree 3, then $χ_c'(G) \leq 11/3$ provided that $G$ does not contain $H_1$ or $H_2$ as a subgraph, where $H_1$ and $H_2$ are obtained by subdividing one edge of $K_2^3$ (the graph with three parallel edges between two vertices) and $K_4$, respectively. As $χ_c'(H_1) = χ_c'(H_2) = 4$, our result implies that there is no graph $G$ with $11/3 < χ_c'(G) < 4$. It also implies that if $G$ is a 2-edge connected cubic graph, then $χ'(G) \le 11/3$.
For constant $r$ and arbitrary $n$, it was known that in the graph $K_r^n$ any independent set of size close to the maximum is close to some independent set of maximum size. We prove that this statement holds for arbitrary $r$ and $n$.
We determine under which conditions certain natural models of random constraint satisfaction problems have sharp thresholds of satisfiability. These models include graph and hypergraph homomorphism, the $(d,k,t)$-model, and binary constraint satisfaction problems with domain size 3.
We study various SDP formulations for {\sc Vertex Cover} by adding different constraints to the standard formulation. We show that {\sc Vertex Cover} cannot be approximated better than $2-o(1)$ even when we add the so called pentagonal inequality constraints to the standard SDP formulation, en route answering an open question of Karakostas~\cite{Karakostas}. We further show the surprising fact that by strengthening the SDP with the (intractable) requirement that the metric interpretation of the solution is an $\ell_1$ metric, we get an exact relaxation (integrality gap is 1), and on the other hand if the solution is arbitrarily close to being $\ell_1$ embeddable, the integrality gap may be as big as $2-o(1)$. Finally, inspired by the above findings, we use ideas from the integrality gap construction of Charikar \cite{Char02} to provide a family of simple examples for negative type metrics that cannot be embedded into $\ell_1$ with distortion better than $8/7-\eps$. To this end we prove a new isoperimetric inequality for the hypercube.