Explicit constructions of infinite families of MSTD sets

We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/2^{r/2} for some polynomial f(r)). We conclude by generalizing our method to compare linear forms epsilon_1 A + ... + epsilon_n A with epsilon_i in {-1,1}.Comment: Version 2: 14 pages, 1 figure. Includes extensions to ternary forms and a conjecture for general combinations of the form Sum_i epsilon_i A with epsilon_i in {-1,1} (would be a theorem if we could find a set to start the induction in general

Twin inequality for fully contextual quantum correlations

Quantum mechanics exhibits a very peculiar form of contextuality. Identifying and connecting the simplest scenarios in which more general theories can or cannot be more contextual than quantum mechanics is a fundamental step in the quest for the principle that singles out quantum contextuality. The former scenario corresponds to the Klyachko-Can-Binicioglu-Shumovsky (KCBS) inequality. Here we show that there is a simple tight inequality, twin to the KCBS, for which quantum contextuality cannot be outperformed. In a sense, this twin inequality is the simplest tool for recognizing fully contextual quantum correlations.Comment: REVTeX4, 4 pages, 1 figur

The fractional chromatic number of triangle-free subcubic graphs

Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909)

On local structures of cubicity 2 graphs

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation

Fractional isomorphism of graphs

AbstractLet the adjacency matrices of graphs G and H be A and B. These graphs are isomorphic provided there is a permutation matrix P with AP=PB, or equivalently, A=PBPT. If we relax the requirement that P be a permutation matrix, and, instead, require P only to be doubly stochastic, we arrive at two new equivalence relations on graphs: linear fractional isomorphism (when we relax AP=PB) and quadratic fractional isomorphism (when we relax A=PBPT). Further, if we allow the two instances of P in A=PBPT to be different doubly stochastic matrices, we arrive at the concept of semi-isomorphism.We present necessary and sufficient conditions for graphs to be linearly fractionally isomorphic, we prove that quadratic fractional isomorphism is the same as isomorphism and we relate semi-isomorphism to isomorphism of bipartite graphs